In [1], the notion of an $a$-framed bordism was introduced, where $a$ is the Schubert symbol. The problem of calculating the groups of stably $a$-bordant closed $a$-framed submanifolds of Euclidean spaces reduces to the problem of calculating the stable homotopy groups of some Thom spaces. In the present paper, we study the cohomology rings modulo 2 of these spaces. As an application in one case, a description is given of the corresponding graded cohomology group as a module over the Steenrod algebra A2.