  Let $E$ be the natural representation of the special linear group
$\mathrm{SL}_2(K)$ over an arbitrary field $K$. We use the two dual
constructions of the symmetric power when $K$ has prime characteristic to
construct an explicit isomorphism $\mathrm{Sym}_m \mathrm{Sym}^\ell E \cong
\mathrm{Sym}_\ell \mathrm{Sym}^m E$. This generalises Hermite reciprocity to
arbitrary fields. We prove a similar explicit generalisation of the classical
Wronskian isomorphism, namely $\mathrm{Sym}_m \mathrm{Sym}^\ell E \cong
\bigwedge^m \mathrm{Sym}^{\ell+m-1} E$. We also generalise a result first
proved by King, by showing that if $\nabla^\lambda$ is the Schur functor for
the partition $\lambda$ and $\lambda^\circ$ is the complement of $\lambda$ in a
rectangle with $\ell+1$ rows, then $\nabla^\lambda \mathrm{Sym}^\ell E \cong
\nabla^{\lambda^\circ} \mathrm{Sym}_\ell E$. To illustrate that the existence
of such `plethystic isomorphisms' is far from obvious, we end by proving that
the generalisation $\nabla^\lambda \mathrm{Sym}^\ell E \cong \nabla^{\lambda'}
\mathrm{Sym}^{\ell + \ell(\lambda') - \ell(\lambda)}E$ of the Wronskian
isomorphism, known to hold for a large class of partitions over the complex
field, does not generalise to fields of prime characteristic, even after
considering all possible dualities.
