It is known that the lattice structure of an interval of varieties of involution semigroups can be very different from that of its reduct interval of varieties of semigroups, but it is uncertain how wide this difference can be. The present article exhibits intervals to show that the difference can be between being a two-element chain and having an infinite chain. These intervals are bounded by varieties generated by an involution semigroup of order between four and six. One of the involution semigroups of order four turns out to be a smallest involution semigroup to generate a non-Specht variety. In contrast, it is long known that every semigroup of order up to four generates a Specht variety.