This is a general question I have and not related to any specific problem. Suppose we have two groups $G, H$ with respective actions $A: G \times X \rightarrow X$ and $B: H \times X \rightarrow X$ on some set $X$. If $G \cong H$, does this imply any interesting relations between $A$ and $B$? I know that $A$ and $B$ need not be related with regards to primitivity, blocks etc.
My intuition is no, since $A$ could just be the trivial group action. If this is the case, does adding any constraints (say faithfulness) give us more interesting results? Thanks.