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• For $\aleph_2$: $1$, $3$, $5$, $6$, $7$, $12$, or $19$ models.
• For $\aleph_3$: $1$, $4$, $7$, $10$, $13$, $22$, or $37$ models.

• For $\aleph_2$: $1$, $3$, $5$, $6$, $7$, $12$, or $19$ models. (Note that this list is only complete for the numbers less than $10$.)
• For $\aleph_3$: $1$, $4$, $7$, $10$, $13$, $22$, or $37$ models. (Note that this list is only complete for the numbers less than $20$.)

Let $T$ be a complete countable first-order theory. I will write $I(T,\kappa)$ for the number of models of $T$ of cardinality $\kappa$ up to isomorphism. The function $I(T,-)$ is called the spectrum of $T$. The possible spectra are completely understood (for uncountable $\kappa$), see The uncountable spectra of countable theories by Hart, Hrushovki, and Laskowki). For background, I'll sketch what's going on in the cases when some $I(T,\kappa)$ is finite, and then I'll show that the set $X$ is computable.

The question James raised in the comments, about whether the numbers that occur as $p_G(k)$ or $q_G(k)$ can be characterized in some nicer way that doesn't involve enumerating all finite groups, is also interesting.

Let $T$ be a complete countable first-order theory. I will write $I(T,\kappa)$ for the number of models of $T$ of cardinality $\kappa$ up to isomorphism. The function $I(T,-)$ is called the spectrum of $T$. The possible spectra are completely understood (for uncountable $\kappa$), see The uncountable spectra of countable theories by Hart, Hrushovski, and Laskowski). For background, I'll sketch what's going on in the cases when some $I(T,\kappa)$ is finite, and then I'll show that the set $X$ is computable.

The question James raised in the comments, about whether the numbers that occur as $p_G(k)$ or $q_G(k)$ can be characterized in some nicer way that doesn't involve enumerating all finite permutation groups, is also interesting.

Now the models of $T$ of cardinality less than $\aleph_k$ can be classified by a $d$-tuple of infinite cardinals $< \aleph_k$, which we can code by an element of $k^d$. But there's an additional hitch: the automorphism group of the prime model might act non-trivially on the $d$ strongly minimal sets. So associated to $T$, we have a subgroup $G$ of $S_d$, and the number of models of $T$ of cardinality less than $\aleph_k$ is equal to the number of orbits of the natural action of $G$ on $k^d$ (by permuting the indices of the $d$-tuple from $k$). Further, given any subgroup $G$ of $S_d$, we can cook up a theory whose models are classified in this way.

For example, since $q_{S_4}(2) = {2+4-1\choose 2} = {5 \choose 2} = 10$, and $q_{S_4}(3) = {3+4-1\choose 3} = {6 \choose 3} = 20$, to determine which numbers less than $10$ occur as $I(T,\aleph_2)$ for some $T$, or to determine which numbers less than $20$ occur as $I(T,\aleph_3)$ for some $T$, we only have to consider subgroups of $S_3$.

Now the models of $T$ of cardinality less than $\aleph_k$ can be classified by a $d$-tuple of infinite cardinals $< \aleph_k$, which we can code by an element of $k^d$. But there's an additional hitch: the automorphism group of the prime model might act non-trivially on the $d$ strongly minimal sets. So associated to $T$, we have a subgroup $G$ of $S_d$, and the number of models of $T$ of cardinality less than $\aleph_k$ is equal to the number of orbits of the natural action of $G$ on $k^d$ (by permuting the indices of the $d$-tuple from $k$). Further, given any subgroup $G$ of $S_d$, we can cook up a theory whose models are classified in this way: basically, we find a finite structure $A$ of size $d$ whose automorphism group is $G$, and then consider the theory of an equivalence relation with $d$ classes, each of which is infinite, and equip the quotient with the structure of $A$.

For example, since $q_{S_d}(1)={1+d-1\choose 1}=d$, every natural number $m$ can occur as $I(T,\aleph_1)$ for some $T$ (explicitly, the theory of an equivalence relation with $m$ classes, all of which are infinite - the models of size $\aleph_1$ are determined up to isomorphism by how many classes are uncountable)

Since $q_{S_4}(2) = {2+4-1\choose 2} = {5 \choose 2} = 10$, and $q_{S_4}(3) = {3+4-1\choose 3} = {6 \choose 3} = 20$, to determine which numbers less than $10$ occur as $I(T,\aleph_2)$ for some $T$, or to determine which numbers less than $20$ occur as $I(T,\aleph_3)$ for some $T$, we only have to consider subgroups of $S_3$.