Add custom page editor buttons to a FieldRenderer's edit frame

Add custom page editor buttons to a FieldRenderer's edit frame

If I have a <sc:FieldRenderer> or related control in a sublayout, and I
wrap it in a <sc:Link> control, the FieldRenderer will have an additional
button to edit the link field, rather than a second wrapping EditFrame.
Is it possible for other buttons to be added to the built-in field
renderer controls in the same manner rather than using an <sc:EditFrame>
control?
For example, if I used the following markup:
Good:
<div class="example">
<sc:Text runat="server" ID="Text" Field="Text" />
</div>
Is it possible to add custom buttons to the edit frame shown in the page
editor for the Text control, without needing to rely on an EditFrame
control?
Bad:
<div class="example">
<sc:EditFrame runat="server" ID="Edit">
<sc:Text runat="server" ID="Text" Field="Text" />
</sc:EditFrame>
</div>