  In this article the study of the Prime Graph Question for the integral group
ring of almost simple groups which have an order divisible by exactly $4$
different primes is continued. We provide more details on the recently
developed "lattice method" which involves the calculation of
Littlewood-Richardson coefficients. We apply the method obtaining results
complementary to those previously obtained using the HeLP-method. In particular
the "lattice method" is applied to infinite series of groups for the first
time. We also prove the Zassenhaus Conjecture for four more simple groups.
Furthermore we show that the Prime Graph Question has a positive answer around
the vertex $3$ provided the Sylow $3$-subgroup is of order $3$.
