How many stripes can you paint on a $2$-group of fixed size?
A group of order $2^ap^b$ is solvable, by Burnside's theorem, so its chief factors are either abelian $2$-groups or abelian $p$-groups. Such a group is a zebra group if its chief factors in any chief series alternate between the $2$ and the $p$. In other words, if you take a chain of normal subgroups $$1 = N_1 ⊲ N_2 ⊲ \cdots ⊲ N_n = G$$ of maximal length, then the quotient groups $N_{i+1}/N_i$ alternate between being abelian $2$-groups (the stripes) and abelian $p$-groups (the background).
If we fix $a$, say $a=8$, then how many stripes can a zebra group have?
Obviously no more than $8$ $2$-stripes, but with a little work one can see that it can have no more than $4$ $2$-stripes. Unfortunately, I'm having trouble getting even $3$ stripes.