Natural maps between Schur functors: understanding the image

Question

Let $V$ be a finite dimensional representation of symmetric group $\mathbb{S}_n.$ Consider a natural map $$\pi \colon \Lambda^2 V \otimes \Lambda^2 V \longrightarrow \Lambda^4 V.$$ Let $[\Lambda^2 V]_{\lambda}$ be an isotypical component representation $\Lambda^2 V$ corresponding to some Young diagram $\lambda.$ Is there a simple way to compute the character of representation $ \pi([\Lambda^2 V]_{\lambda} \otimes \Lambda^2 V)? $ Of course, this question can be easily generalized to other natural morphism between various compositions of Schur functors.