From Sela's proof of Tarski's conjecture we know that the surface groups (i.e. fundamental group of a closed surface of genus $\geq 2$) and free (non-abelian) groups have the same first order theory. We also know that the inclusion of a free non-abelian group into another as a free factor is an elementary embedding. Now any subgroup $H$ of a surface group $G$ is either a surface group (if the index is finite) or a free group (if the index is infinite). Hence the elementary theory of $H$ is equivalent to that of $G$. On the other hand not every inclusion of a subgroup of the surface group is an elementary embedding. A necessary condition is provided in Theorem 1.4 of this paper.
Q) Is there a sufficient condition for the inclusion to be an elementary embedding?
Q) Can we completely classify the subgroups of surface groups for which the inclusion is an elementary embedding?
I am a newcomer in this area and I am currently reading Sela's proof of the above theorems therefore any suggestion about papers or references regarding the problems will be extremely helpful. Thanks in advance.