TITLE: What can we say about the kernel of $\phi: F_n \rightarrow S_k$
QUESTION [8 upvotes]: Let $F_n$ denote the free group on $n$ generators and let $S_k$ denote the symmetric group on the integers $\{1,\dots, k\}$, and the action of homomorphism $\phi$ (as given in the title) on the generators of $F_n$ be known. Let $K_{\phi}$ denote the kernel.
My question is two-fold.

I've had GAP spit out a couple dozen random examples of such homomorphisms/kernels and in every case so far $K_{\phi}$ appears to be of infinite generation. Is this always the case? I can't imagine how it couldn't be if $\phi$ is chosen to be nontrivial (my reasoning is that, even if $\phi$ kills off all but one generator, so that $K_{\phi}$ is a subset of an infinite cyclic subgroup of $F_n$, $K_{\phi}$ is still not finitely generated, and if $\phi$ kills off fewer-than-all-but-one generators the complexity only increases) but I also can't see how I would begin a proof, unless my reasoning in the previous parenthetical is the proof strategy. (Supposing, though, that there is a counterexample, is there a way to tell from the presentation of $\phi$ whether or not $K_{\phi}$ is finitely-presented?)
Can anything be said about the "combinatorics" of the embedding $\iota: K_{\phi} \hookrightarrow F_n$? (To clarify what I mean by this: Of course, if all the groups that arise as kernels in this way are on infinite generators then any two are isomorphic by Nielsen-Schrier, but even if two such arise from the same $F_n$, they're clearly not necessarily the same subgroup.)

Edit: I seem to have forgotten or never learned something important--As James points out in a comment below, $K_{\phi}$ is a finite-index subgroup of a finitely-generated group, so it's necessarily finitely-generated by what I just learned is called Schreier's lemma. With that in mind, I think it's sensible to revise the first part of my question to "What can we say about a generating set of $K_{\phi}$ given what we know about $\phi$?" The second part of the question remains a question for the time being.
I've given this the reference request tag in case there's a standard source on these topics that I should be familiar with. If anything is unclear, please post a comment and I'll try to fix it.

REPLY [6 votes]: As the following shows, GAP can work out generators of the kernel. Perhaps you were put off by the fact that it said "no generators known". That just means that it has not calculated them yet.
The other possibility is that the example you gave it was too large. If you tried mapping $F_{20}$ to $S_{20}$ for example, then $K$ would have far too many generators for GAP to want to start calculating and storing them. That is the main reason why it does not try to calculate the generators before you ask it to.
 gap> F:=FreeGroup(2);;
 gap> S:=SymmetricGroup(5);;
 gap> f := GroupHomomorphismByImages(F, S, [F.1,F.2], [(1,2),(1,2,3,4,5)]);;
 gap> K:=Kernel(f);
   Group(<free, no generators known>)
 gap> GK:=GeneratorsOfGroup(K);;
 gap> Length(GK);
  121