Simple search and radio button group

Hello listers,
maybe somebody can point me in the right direction. On a list page with simple search, I placed a radio button group next to the search field

(I did this via "insert snippet" in page designer).

The button group shows up, any element can be selected - but, how do I manage to submit the

value of the selected radio button together with the text in the search field when the search button is clicked, e.g.

instead of

TIA and regards