Length-preserving monomorphisms for some algebras of operations

Let $p$ be an odd prime. Investigating the existence of a fractal structure for the universal Steenrod algebra $mathcal Q(p)$ and the Lambda algebra leads to determine the group of their length-preserving automorhisms. Contrarily to the $p=2$ case, no length-preserving strict monomorphism turns out to exist, and this makes reasonable to conjecture that the algebras above do not contain proper subalgebras isomorphic to themselves.