Let $ψ$ be a sentence in the counting monadic second-order logic of matroids and let $\mathbb{F}$ be a finite field. Hliněný's Theorem says that we can test whether $\mathbb{F}$-representable matroids satisfy $ψ$ 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 Hliněný'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.