  Let $M$ be a compact surface with boundary. We are interested in the question
of how a group action on $M$ permutes a finite invariant set $X \subset
int(M)$. More precisely, how the algebraic properties of the induced group of
permutations of a finite invariant set affects the dynamical properties of the
group. Our main result shows that in many circumstances if the induced
permutation group is not solvable then among the homeomorphisms in the group
there must be one with a pseudo-Anosov component. We formulate this in terms of
the mapping class group relative to the finite set and show the stronger result
that in many circumstances (e.g. if $\partial M \ne \emptyset$) this mapping
class group is itself solvable if it has no elements with pseudo-Anosov
components.
