(I have posted this question in Math Stack Exchange, only to have received no answer.)

It is well known that a semidirect product of two cyclic groups $C_m$ and $C_n$ has the form $$ C_m \rtimes_k C_n = \langle x,y \mid x^m = y^n = 1,\, yxy^{-1} = x^k \rangle, $$ for some $k^n \equiv 1\pmod m$.

A question that seems quite elementary and yet to which I have not found a satisfactory answer is: what $k$'s give isomorphic semidirect products? Clearly, if $\gcd(r,n)=1$, then $C_m\rtimes_k C_n$ is isomorphic to $C_m\rtimes_{k^r} C_n$ via the mapping $(x,y)\mapsto(x,y^{r^{-1}\ \mathrm{mod}\ n})$. Is the converse also true, that is, if $C_m\rtimes_k C_n$ is isomorphic to $C_m\rtimes_{k'} C_n$, then $k'\equiv k^r\pmod m$ for some $\gcd(r,n)=1$? If so, then we can easily deduce the number of nonisomorphic semidirect products of $C_m$ and $C_n$.

I strongly believe that this is true, but using brute force to verify it seems infeasible. Any help/reference appreciated.