  An 'induced restriction' of a simple binary matroid $M$ is a restriction
$M|F$, where $F$ is a flat of $M$. We consider the class $\mathcal{M}$ of all
simple binary matroids $M$ containing neither a free matroid on three elements
(which we call a 'claw'), nor a Fano plane as an induced restriction. We give
an exact structure theorem for this class; two of its consequences are that the
matroids in $\mathcal{M}$ have unbounded critical number, while the matroids in
$\mathcal{M}$ not containing the clique $M(K_5)$ as an induced restriction have
critical number at most $2$.
