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.
(see above) as the initial
pre-fusion map.
-class into two (or more) parts and goto 2., or
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].