Finding differents set of generators of a group given an initial set

Question

Suppose we have a non abelian finite group $G$ which is generated by the irredundant set $S=\{g_1,g_2,\cdots, g_n\}$. How can we find other irredundant sets of generators of $G$ using $S$? By irredundant sets of generators I mean generators sets in which none element can be expressed as a product of the others.

Answer 1

Take any automorphism (or anti-automorphism) $\phi$ of your group. Then $\phi(g_1),...,\phi(g_n)$ is a generator too.

Example. $\phi(g)=g^{-1}$, or $\phi(g)=h^{-1}gh$ for some $h\in G$.

Less generic examples might depend on the generated group. There might be not much left to do if $G$ is the free group (but I am not sure here).