The small index property of automorphism groups of ab-initio generic structures

Suppose $M$ is a countable ab-initio (uncollapsed) generic structure which is obtained from a pre-dimension function with rational coefficients. We show that if $H$ is a subgroup of $\mbox{Aut}\left(M\right)$ with $\left[\mbox{Aut}\left(M\right):H\right]<2^{\aleph_{0}}$, then there exists a finite s