Free idempotent generated semigroups and endomorphism monoids of free G-acts

We say that the rank of an element of EndFn(G) is the minimal number of (free) generators in its image. Let ε=ε2∈EndFn(G). For rather straightforward reasons it is known that if rankε=n−1 (respectively, n), then the maximal subgroup of IG(E) containing ε is free (respectively, trivial). We show that if rankε=r where 1≤r≤n−2, then the maximal subgroup of IG(E) containing ε is isomorphic to that in EndFn(G) and hence to G≀Sr, where Sr is the symmetric group on r elements. We have previously shown this result in the case r=1; however, for higher rank, a more sophisticated approach is needed. Our current proof subsumes the case r=1 and thus provides another approach to showing that any group occurs as the maximal subgroup of some IG(E). On the other hand, varying r again and taking G to be trivial, we obtain an alternative proof of the recent result of Gray and Ruškuc for the biordered set of idempotents of Tn.