Generators for a subgroup of a free group

Question

Let $F$ be a finitely generated free group and $G$ a finite group and $\phi :F\to G$ some homomorphism. Is there a method to calculate generators for the kernel of $\phi$. Note I'm looking for actual generators not a presentation. If you can add a reference it would be even better.

Answer 1

A way to do this is to look into the proof of the Nielsen-Schreier theorem. Let $X$ be the set of free generators of $F$, and let $U$ be a Schreier transversal for $\ker \phi$. Then construct the set $A = \{ u x \overline{u x}^{-1} \mid u \in U, x \in X^{\pm 1} \}$. It turns out that the set $A$ generates $\ker \phi$. This generating set can be reduced to a minimal one if you require that $ux \notin U$. The obtained subset of $A$ freely generates $\ker \phi$. For all the details, see the book D. L. Johnson: Presentations of Groups, Chapter 2.