On expansions of non-abelian free groups by cosets of a finite index subgroup

Let $F$ be a finitely generated non-abelian free group and $Q$ a finite quotient. Denote by $L_Q$ the language obtained by adding unary predicates $P_q$, $q\in Q$ to the language of groups. Using a slight generalization of some of the techniques involved in Zlil Sela's solution to Tarski\'s problem on the elementary theory of non-abelian free groups, we provide a few basic results on the validity of first order entences in the $L_Q$-expansion of $F$ in which every $P_q$ is interpreted as the preimage of $q$ in $F$. In particular we prove an analogous result to Sela's generalization of Merzlyakov's theorem on $\forall\exists$-sentences and show that the positive theory depends only on $Q$ and neither on the rank of $F$ nor the particular quotient map.