The complete program of how to apply the procedures described in the previous section and when to supplement them with additional information about the group is given by the following strategy.

A rough guideline is given by Lemma 4.3: Start with a trivial pre-fusion map. Keep producing stronger pre-fusion maps by splitting and fusing until a fusion map is reached.

- Start with (see above) as the initial pre-fusion map.
- Refine the equivalence relation with
*RefineClassesFrame*(see 4.5). - Apply
*CheckSylowFrame*(see 4.8) and*CheckNormalizerFrame*(see 4.10) in appropriate places. - Apply
*ConcludeFrame*(see 4.6). - Stop if all classes are singletons. Otherwise find a legitimation for
an action of one of the two following kinds.
- Split an -class into two (or more) parts and goto 2., or
- Fuse two (or more) images
*f*(*m*) and goto 4.

Suppose, for example, that it is known (from the list of conjugacy classes of
elements of *G*) that *G* has exactly two conjugacy classes of elements of
order 2. Then *G* has two conjugacy classes of subgroups of order two.
Suppose further that we know a pre-fusion map such that there
exactly two images *f*(*m*) of size two and these two form one -class.
Then, if the equivalence is derived from by splitting that
class in two, then is a stronger pre-fusion map than . Moreover, this splitting will, via *RefineClassesFrame*, have
an effect on all classes of subgroups that contain subgroups of order 2,
depending on the distribution of the two types of subgroups of order two
within them.

Suppose that the intersections of two classes of maximal subgroups are known,
that one can find two classes and where
and are in fact identical as subgroups of *G*, i.e. where
. Suppose further that we know a pre-fusion map such that . Then, if *f*' is derived from *f*
by fusing the images of and , then is a stronger
pre-fusion map than . Moreover, this fusion will, via *
ConcludeFrame*, have an effect on all classes which contain subgroups of
or , because they also must fuse in some way. For examples of
intersections of maximal subgroups (and methods how to determine them) see
e.g. [Komissartschik and Tsaranov 1986].