  Let $\psi$ be a sentence in the counting monadic second-order logic of
matroids and let $\mathbb{F}$ be a finite field. Hlin\v{e}n\'{y}'s Theorem says
that we can test whether $\mathbb{F}$-representable matroids satisfy $\psi$
using an algorithm that is fixed-parameter tractable with respect to
branch-width. In a previous paper we proved there is a similar fixed-parameter
tractable algorithm that can test the members of any efficiently pigeonhole
class. In this sequel we apply results from the first paper and thereby extend
Hlin\v{e}n\'{y}'s Theorem to the classes of fundamental transversal matroids,
lattice path matroids, bicircular matroids, and $H$-gain-graphic matroids, when
$H$ is a finite group. As a consequence, we can obtain a new proof of
Courcelle's Theorem.
