Given a Yang-Mills theory such as $SU(3)$ which has 8 gluons. After we gauge-fix this theory, it no longer has $SU(3)$ guage symmetry. Yet, we still use the group constants and the 8 types of gluons to calculate the Feynman graphs.
Thus the gauge-fixed action still retains some sort of $SU(3)$-ness to it even though it is not a local symmetry any more.
In the gauge-fixed action, what is the name of this $SU(3)$-ness?
It's like we start with a guage theory with group $G$, then gauge-fix it to remove the guage freedoms, but are left with a shadow of the group $G$.
i.e. we could still classify all gauge-fixed Yang-Mills actions by a group $G$.
So what is the relation of the gauge group to the gauge-fixed action?