On groups generated by involutions of a semigroup

An involution on a semigroup S   (or any algebra with an underlying associative binary operation) is a function α:S→Sα:S→S that satisfies α(xy)=α(y)α(x)α(xy)=α(y)α(x) and α(α(x))=xα(α(x))=x for all x,y∈Sx,y∈S. The set I(S)I(S) of all such involutions on S   generates a subgroup C(S)=〈I(S)〉C(S)=〈I(S)〉 of the symmetric group Sym(S)Sym(S) on the set S  . We investigate the groups C(S)C(S) for certain classes of semigroups S  , and also consider the question of which groups are isomorphic to C(S)C(S) for a suitable semigroup S.