To answer your question: Yes, I believe this is achievable using contextual filters.
To be more specific: I believe you'll be able to find a contextual filter "Content: Has taxonomy term ID". It's hard to be able to give you the information on what should go in here without knowing how the pages are constructed. But, for example, if you had pages that had the taxonomy term ID (for the Groups vocabulary) in the url, you could select:
Provide default value > Taxonomy term ID from URL. You may have to use "Raw value from URL" and select 2 under path component because of how your urls are constructed
There's also "Specify validation criteria".
Select "Taxonomy term" from that and you can limit it by vocabulary or even convert taxonomy term names into IDs.
To use a taxonomy term name from the url under "Filter value type" select "Term name" or "Term name converted to Term id" (note, it is much safer to use taxonomy term ids in the url as terms with the same name can not be distinguished). If you are using term names make sure you check "Transform dashes in URL to spaces in term name filter values"
Hope that answers your question :)