  We continue our study of symmetries of a class of little string theories of
A-type, which are engineered by $N$ parallel M5-branes probing a flat
transverse space. Extending the analysis of the companion paper, we discuss the
part of the free energy that is sensitive to the details of the
$\mathfrak{a}_{N-1}$ gauge structure, by computing explicit series expansions
for the cases $N=2,3,4$. Based on these examples, we find a class of functions
that we conjecture to resum whole sectors in the instanton expansion of the
free energy and which combine in a natural manner its modular properties as
well as the gauge symmetry. These functions have previously been introduced in
the literature as the generating functions of multi-divisor sums and in the
case $N=2$ can also be cast into the form of a generalised Eisenstein series.
We use these resummed contributions to the free energy to perform a number of
non-trivial consistency checks for our results.
