  For a prime $p$, we describe a protocol for handling a specific type of
fusion system on a $p$-group by computer. These fusion systems contain all
saturated fusion systems. This framework allows us to computationally determine
whether or not two subgroups are conjugate in the fusion system for example. We
describe a generation procedure for automizers of every subgroup of the
$p$-group. This allows a computational check of saturation. These procedures
have been implemented using MAGMA. We describe a program to search for
saturated fusion systems $\mathcal{F}$ on $p$-groups with $O_p(\mathcal{F})=1$
and $O^p(\mathcal{F})=\mathcal{F}$. Employing these computational methods we
determine all such fusion system on groups of order $p^n$ where $(p,n) \in
\{(3,4),(3,5),(3,6),(3,7),(5,4),(5,5),(5,6),(7,4),(7,5)\}$. This gives the
first complete picture of which groups can support saturated fusion systems on
small $p$-groups of odd order.
