Difficult# Abelian Groups Acting Transitively

ABSALG-BFQQNX

Let $A$ be an abelian group of order $a$ acting transitively on a set $X$ of size $n$. Suppose that $A$ acts faithfully (i.e. for every non-trivial $g \in A$ there exists $x \in X$ such that $g \cdot x \neq x$).

Which of the following are possible values for $n$?