If $G$ acts on $A$ and $A\cong B$, how does $G$ act on $B$?

Question

Let $G$ be a group acting on a vector space $A$. If $T:A\rightarrow B$ is an isomorphism, it seems natural that $G$ should also act on $B$. My guess is that the resulting action $B$ should be defined by $$ g.b:= T(g.T^{-1}(b)). $$ It clearly satisfies $(gh).b=g.h.b$ for any $g,h\in G$. Does this sound correct?