Is this definition about an operation on a group wrong?

Question

Let $(G,+)$ is a group and $G'\neq\varnothing$ $\varphi\colon G\to G'$ is bijective,define an operation $*$ on $G'$ by $g'*h' \colon=\varphi^{-1}(g')+\varphi^{-1}(h')$ $g',h'\in G'$ then $(G,*)$ is a group

The operation on $G'$ should be closed on $G'$,but the definition imply that the $g'*h'$ belongs to $G$, which contradict the defition of group.

Answer 1

Yes, it should be $g'*h'=\phi(\phi^{-1}(g')+\phi^{-1}(h'))$

and $(G', *)$ is a group

Answer 2

Yes, it is wrong. I suspect that who wrote that meant$$g'*h'=\varphi\bigl(\varphi^{-1}(g')+\varphi^{-1}(h')\bigr).$$