Let $f$ be an automorphism of a finite group $G$ generated by the elements $\{g_1, g_2, \cdots, g_i\}$. $f^m$ is the automorphism $f$ composed $m$ times. i.e. $f^m = f \circ f \cdots \circ f (m \,\,times)$.
Suppose for some $k \in \mathbb{Z}^{+}$, the values of $f^k(g_1), f^k(g_2), \cdots , f^k(g_i)$ are computed by one person (say $A$) and these values are transmitted to another person ($B$).
Since $B$ knows only $f^k(g_1), f^k(g_2), \cdots , f^k(g_i)$ values he can not find what the automorphism $f$ is, right?
Please help me to clarify regarding the above idea.
Thanks a lot.
If you know the value of $k$ it might be possible to find out the automorphism. If $|G| = n$, since an automorphism fixes the identity, note that the automorphisms of $G$ are naturally a subgroup of $S_{n-1}$. Hence, it could happen that there is only one element in $S_{n-1}$ satisfying $g^k = \phi$ (where $\phi$ is the automorphism you've been given). For example, in $S_3$, for a given $3$-cycle it has a unique square root.
If this doesn't work, you can look at the group of automorphisms and do a similar analysis but that might be a bit more complicated.