Gaieus beat me to it! 😮
Here my go anyway... It reiterates what he's just said...
The intersection geometry appears in the 'current context'.
So if you have two overlapping groups in the model, you select them and use 'Intersect with selection', then all the new edges are put in the model context.
When you then explode a group it merges with this geometry.
A much easier way is this...
Do the Intersection as above.
Select the new geometry [suggest you use a left>right fence to include just it and omit the groups, but if you accidentally pick something else too you can remove it from the selection by holding down Shift+Ctrl and picking it again; conversely you can add missed things to the selection by holding down Ctrl as you pick those].
Now you have all of the intersected geometry selected.
Edit>Cut.
Edit the group that you want to contain this new geometry [double-click to move into its context]
Edit>PasteInPlace.
The pasted geometry is now added to the group and merges with the existing geometry as appropriate.
Edit the group's contents as desired... and then close its edit session.
Another way that is even easier [provided that only the two groups intersect and there are no other overlaps you want to miss out]... is to initially edit the group where the intersecting new geometry is wanted, and use Edit>SelectAll.
Now use 'Intersect with model' and the new geometry will form inside the group as it is made, merging as appropriate...
Tip:
To see what you are doing when editing something like this, use the View>ComponentEdit settings which can temporarily hide the rest of the model and/or similar components [groups and components are treated the same way for this 'toggle'] as you edit. It's useful to see what you are doing, but of course sometimes seeing the rest of the model outside of the edit context is useful as you can still snap to it etc...