If $B$ is a subalgebra of $A$, you can ask whether the $B$-module structure on $B$ can be extended to give an $A$-module structure on $B$.

W H Lin, in his 1973 PhD thesis at Northwestern, showed that the only Hopf subalgebras of the mod 2 Steenrod algebra for which this can be done are the algebras $A(n)$ — this is the algebra generated by $\text{Sq}^{2^i}$ for $i\leq n$. Are there any electronic copies of the thesis, or at least this particular proof, available?