Automorphisms groups of countable recursively saturated models

Generic automorphisms and the small index property

Lascar showed that countable arithmetically saturated models of $PA$ have generic automorphisms and used them to show that automorphism groups of these models have the small index property.

Do automorphism groups of countable recursively saturated, but nor arithmetically saturated models of $PA$ have generic automorphisms? Do they have the small index property.

It was shown by Kossak and Schmerl that every open subgroup of the automorphism group of a countable recursively saturated model of $PA$ extends to a maximal subgroup. Is the same true for countable models which are recursively saturated, but not arithmetically saturated?

References:

Lascar, Daniel, The small index property and recursively saturated models of Peano arithmetic. Automorphisms of first-order structures, 281–292, Oxford Sci. Publ., Oxford Univ. Press, New York, 1994.

Kossak, Roman; Kotlarski, Henryk; Schmerl, James H. On maximal subgroups of the automorphism group of a countable recursively saturated model of PA. Ann. Pure Appl. Logic 65 (1993), no. 2, 125–148

How many automorphism groups are there?

By a result of Kossak and Schmerl, if $M$ and $N$ are countable arithmetically saturated models of the same completion of $PA$, then $M\cong N$ iff ${\rm Aut}(M)\cong {\rm Aut}(N)$. It is open whether the same is true for recursively saturated models which are not arithmetically saturated. It is even open whether there are nonisomorphic countable recursively saturated models whose automorphism groups are isomorphic.

Nurkhaidarov proved that there is a set $\mathcal T$ of completions of $PA$, such that $|{\mathcal T}|=4$ and for all $T, T'\in {\mathcal T}$, if $T\not=T'$, and $M\models T$ and $M'\models T'$ are countable and arithmetically saturated, then ${\rm Aut}(M)\not\cong{\rm Aut}(M')$. Surely, there must be more than four such completions. How many are there?

References:

Kossak, Roman; Schmerl, James H. The automorphism group of an arithmetically saturated model of Peano arithmetic.J. London Math. Soc. (2) 52 (1995), no. 2, 235–244.

Nurkhaidarov, Ermek S. Automorphism groups of arithmetically saturated models. J. Symbolic Logic 71 (2006), no. 1, 203–216.