Let be a pre-fusion map on .

Define, for subgroups *U* and *A* of *G*, the set as the set of all subgroups *B* of *U* which lie in the
-class of *A*. Denote by its size. This
number is determined as follows. If *U* is a subgroup of the maximal
subgroup of *G*, then (by Proposition 1.4)

where the sums range over all representatives *B* of conjugacy classes of
with .

This property can be used to detect elements of which are not
conjugate in *G*.

A procedure which systematically applies Corollary 4.5 to
in order to produce the strongest possible pre-fusion map from will be called *RefineClassesFrame*.

Comparing the sets *S*(*U*, *A*) for conjugate subgroups *U* leads to even more
detailed insights.

Let and let . Let

be the image of *S*(*U*, *A*) under *f*. Note that by the definition of *S*(*U*,
*A*) all elements lie in the same -class. To each
we associate a number defined by

where the sum runs over the conjugacy classes of subgroups [*B*] of
with . Conjugation in *G* partitions the set *F*(*U*, *A*) such that
the sum of the corresponding to one part gives the number of
*G*-conjugate subgroups of *U* of a certain type.

This last result appears to be not as explicit as one might wish. In many
cases, however, it is possible to explicitly determine the partitions and
thus a new map *f*'.

If, for example, contains only one element, then,
regardless of the size of *F*(*U*', *A*), the partition will be trivial and *f*'
can be defined as *f*'(*s*') = *s* for each *s*' such that .

A procedure which systematically searches for places where
Corollary 4.6 can be applied in order to produce the strongest
possible pre-fusion map from will be called *
ConcludeFrame*.