adding trigger to listbox grouped items

Next example shows grouped items when your mouse is over the group, hope this is what you requested:

<Grid>
    <Grid.Resources>
        <XmlDataProvider x:Key="Company" XPath="/Company">
            <x:XData>
                <Company xmlns="">
                    <Person Name="Jack" Role="CEO"/>
                    <Person Name="Tim" Role="PL" />
                    <Person Name="Jil" Role="PL" />
                    <Person Name="Jimmy" Role="PM" />
                    <Person Name="Joy" Role="PM" />
                    <Person Name="Jim" Role="PL" />
                    <Person Name="Jack" Role="PM" />
                </Company>
            </x:XData>
        </XmlDataProvider>

        <DataTemplate x:Key="categoryTemplate">
            <TextBlock Text="{Binding Path=Name}" FontWeight="Bold" Background="Gray" Margin="0,5,0,0"/>
        </DataTemplate>

        <DataTemplate x:Key="template">
            <TextBlock Text="{Binding XPath=@Name}"/>
        </DataTemplate>

        <Style x:Key="ContainerStyle" TargetType="{x:Type GroupItem}">
            <Setter Property="Margin" Value="0,0,0,5"/>
            <Setter Property="Template">
                <Setter.Value>
                    <ControlTemplate TargetType="{x:Type GroupItem}">
                        <StackPanel x:Name="Container">
                            <ContentPresenter Grid.Row="0"
                                Content="{TemplateBinding ContentControl.Content}"
                                ContentTemplate="{TemplateBinding ContentControl.ContentTemplate}"
                                ContentStringFormat="{TemplateBinding ContentControl.ContentStringFormat}" />
                            <ItemsPresenter x:Name="Items" Grid.Row="1"
                                Visibility="Collapsed"
                                Margin="5,0,0,0" />
                        </StackPanel>
                        <ControlTemplate.Triggers>
                            <Trigger Property="IsMouseOver" SourceName="Container" Value="True">
                                <Setter TargetName="Items" Property="Visibility" Value="Visible"/>
                            </Trigger>
                        </ControlTemplate.Triggers>
                    </ControlTemplate>
                </Setter.Value>
            </Setter>
        </Style>

        <CollectionViewSource x:Key="cvs" Source="{Binding Source={StaticResource Company},XPath=Person}">
            <CollectionViewSource.GroupDescriptions>
                <PropertyGroupDescription PropertyName="@Role"/>
            </CollectionViewSource.GroupDescriptions>
        </CollectionViewSource>
    </Grid.Resources>


    <ListBox Name="lst"  ItemTemplate="{StaticResource template}" ItemsSource="{Binding Source={StaticResource cvs}}">
        <ListBox.GroupStyle>
            <GroupStyle HeaderTemplate="{StaticResource categoryTemplate}" ContainerStyle="{StaticResource ContainerStyle}" />
        </ListBox.GroupStyle>
    </ListBox>
</Grid>

Added after question modified:
Just use right binding / property in triggers, know you visual tree & expected result.
Next code will show buttons when you mouse is over the groupitem

<Grid>
       <Grid.Resources>
           <XmlDataProvider x:Key="Company" XPath="/Company">
               <x:XData>
                   <Company xmlns="">
                       <Person Name="Jack" Role="CEO"/>
                       <Person Name="Tim" Role="PL" />
                       <Person Name="Jil" Role="PL" />
                       <Person Name="Jimmy" Role="PM" />
                       <Person Name="Joy" Role="PM" />
                       <Person Name="Jim" Role="PL" />
                       <Person Name="Jack" Role="PM" />
                   </Company>
               </x:XData>
           </XmlDataProvider>

           <DataTemplate x:Key="categoryTemplate">
               <TextBlock Text="{Binding Path=Name}" FontWeight="Bold" Background="Gray" Margin="0,5,0,0"/>
           </DataTemplate>

           <DataTemplate x:Key="template">
               <StackPanel>
                   <TextBlock Text="{Binding XPath=@Name}"/>
                   <Button x:Name="Button" Content="Button" Visibility="Hidden"/>
               </StackPanel>
               <DataTemplate.Triggers>
                   <DataTrigger Binding="{Binding Path=IsMouseOver, RelativeSource={RelativeSource AncestorType={x:Type GroupItem}}}" Value="true">
                       <Setter Property="Visibility" Value="Visible" TargetName="Button"/>
                   </DataTrigger>
               </DataTemplate.Triggers>
           </DataTemplate>

           <CollectionViewSource x:Key="cvs" Source="{Binding Source={StaticResource Company},XPath=Person}">
               <CollectionViewSource.GroupDescriptions>
                   <PropertyGroupDescription PropertyName="@Role"/>
               </CollectionViewSource.GroupDescriptions>
           </CollectionViewSource>
       </Grid.Resources>


       <ListBox Name="lst"  ItemTemplate="{StaticResource template}" ItemsSource="{Binding Source={StaticResource cvs}}">
           <ListBox.GroupStyle>
               <GroupStyle HeaderTemplate="{StaticResource categoryTemplate}" />
           </ListBox.GroupStyle>
       </ListBox>
   </Grid>

Answer for question v5:
If problem still doesnt solved, there is some tips for you:
-Add some property to ExecutionControl, indicating buttons visibility, IsButtonsVisible for example...
-Bind inner buttons visibility to that property, use ValueConverters if needed (bool to visibility, etc)
-Modify your ExecutionComponentTemplate datatemplate, set IsButtonsVisible property to false by default
-Modify trigger in ExecutionComponentTemplate, change setter in trigger to set IsButtonsVisible to true when parent GroupItem IsMouseOver is true.