  Let $p$ be a prime, $G$ a finite $\mathcal{K}_p$-group, $S$ a Sylow
$p$-subgroup of $G$ and $Q$ be a large subgroup of $G$ in $S$. The aim of the
Local Structure Theorem is to provide structural information about subgroups
$L$ with $S \leq L$, $O_p(L) \not= 1$ and $L \not\leq N_G(Q)$. There is,
however, one configuration where no structural information about $L$ can be
given using the methods in the proof of the Local Structure Theorem. In this
paper we show that for $p=2$ this hypothetical configuration cannot occur. We
anticipate that our theorem will be used in the programme to revise the
classification of the finite simple groups.
