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.