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)$.