  Let $\mathcal{N}$ be a set of matroids. A matroid $M$ is strictly
$\mathcal{N}$-fragile if $M$ has a member of $\mathcal{N}$ as minor and, for
all $e \in E(M)$, at least one of $M\backslash e$ and $M/e$ has no minor in
$\mathcal{N}$. In this paper we give a structural description of the strictly
$\{U_{2,5},U_{3,5}\}$-fragile matroids that have six inequivalent
representations over $\mathrm{GF}(5)$. Roughly speaking, these matroids fall
into two classes. The matroids without an $\{X_8, Y_8, Y_8^{*}\}$-minor are
constructed, up to duality, from one of two matroids by gluing wheels onto
specified triangles. On the other hand, those matroids with an $\{X_8, Y_8,
Y_8^{*}\}$-minor can be constructed from a matroid in $\{X_8, Y_8, Y_8^{*}\}$
by repeated application of elementary operations, and are shown to have path
width 3.
  The characterization presented here will be crucial in finding the explicit
list of excluded minors for two classes of matroids: the Hydra-5-representable
matroids and the 2-regular matroids.
