A new generalization of Hermite's reciprocity law

Given a partition $\lambda$ of $n$, the {\it Schur functor} $\mathbb{S}_\lambda$ associates to any complex vector space $V$, a subspace $\mathbb{S}_\lambda(V)$ of $V^{\otimes n}$. Hermite's reciprocity law, in terms of the Schur functor, states that $\mathbb{S}_{(p)}\left(\mathbb{S}_{(q)}(\mathbb{C}^2)\right)\simeq \mathbb{S}_{(q)}\left(\mathbb{S}_{(p)}(\mathbb{C}^2)\right).$ We extend this identity to many other identities of the type $\mathbb{S}_{\lambda}\left(\mathbb{S}_{\delta}(\mathbb{C}^2)\right)\simeq \mathbb{S}_{\mu}\left(\mathbb{S}_{\epsilon}(\mathbb{C}^2)\right)$.