The group fixed by a family of endomorphisms of a surface group

For a closed surface S with χ(S)<0, we show that the fixed subgroup of a family B of endomorphisms of π1(S) has rankFixB≤rankπ1(S). In particular, if B contains a non-epimorphic endomorphism, then rankFixB≤12rankπ1(S). We also show that geometric subgroups of π1(S) are inert, and hence the fixed subgroup of a family of epimorphisms of π1(S) is also inert.