  The aim of this paper is to study the resurgent transseries structure of the
inhomogeneous and $q$-deformed Painlev\'e II equations. Appearing in a variety
of physical systems we here focus on their description of $(2,4)$-super minimal
string theory with either D-branes or RR-flux backgrounds. In this context they
appear as double scaled string equations of matrix models, and we relate the
resurgent transseries structures appearing in this way with explicit matrix
model computations. The main body of the paper is focused on studying the
transseries structure of these equations as well as the corresponding
resurgence analyses. Concretely, the aim will be to give a recursion relation
for the transseries sectors and obtain the non-perturbative transmonomials --
{\it i.e.}, the instanton actions of the systems. From the resurgence point of
view, the goal is to obtain Stokes data. These encode how the transseries
parameters jump at the Stokes lines when turning around the complex plane in
order to produce a global transseries solution. The main result will be a
conjectured form for the transition functions of these transseries parameters.
We explore how these equations are related to each other via the Miura map. In
particular, we focus on how their resurgent properties can be translated into
each other. We study the special solutions of the inhomogeneous Painlev\'e II
equation and how these might be encoded in the transseries parameters.
Specifically, we have a discussion on the Hastings--McLeod solution and some
results on special function solutions. Finally, we discuss our results in the
context of the matrix model and the (2,4)-minimal superstring theory.
