This is an analogue of MathOverflow question #138148. Indeed it is so analogous that I wrote the following by copypasting said question and making the necessary changes.

For every commutative ring $A$, let $\mathbf{Symm}_A$ be the ring of symmetric functions over $A$. Let $\mathbf{Symm}$ without a subscript denote $\mathbf{Symm}_{\mathbb{Z}}$.

We can define a bilinear map $\diamondsuit : \mathbf{Symm}_{\mathbb{Q}} \times \mathbf{Symm}_{\mathbb{Q}} \to \mathbf{Symm}_{\mathbb{Q}}$ by setting

$p_{\lambda} \diamondsuit p_{\mu} = \prod\limits_{i\geq 1,\ j\geq 1} p_{\gcd\left(\lambda_i,\mu_j\right)}^{\operatorname{lcm}\left(\lambda_i,\mu_j\right)}$

for any two partitions $\lambda = \left(\lambda_1,\lambda_2,\lambda_3,...\right)$ and $\mu = \left(\mu_1,\mu_2,\mu_3,...\right)$. Here, we are writing $\diamondsuit$ as an infix operator (that is, $a\diamondsuit b$ means $\diamondsuit\left(a,b\right)$), and $p_\nu$ means the $\nu$-power sum symmetric function (defined as $p_{\nu_1} p_{\nu_2} \cdots p_{\nu_k}$, where $\nu$ is written in the form $\left(\nu_1, \nu_2, \ldots, \nu_k\right)$ with all $\nu_i$ positive).

This bilinear map $\diamondsuit$ differs from the arithmetic product $\boxdot$ of MathOverflow question #138148 only in that $\gcd$ and $\operatorname{lcm}$ are switched. Notably, $\diamondsuit$ has no neutral element, whereas $\boxdot$ has $p_1$ as its neutral element. Another difference between $\diamondsuit$ and $\boxdot$ is the lack of Schur-positivity for $\diamondsuit$ (for example, $s_{\left(2,1,1\right)} \diamondsuit s_{\left(2,1,1\right)}$ has both positive and negative coefficients in the Schur basis).

Conjecture: The map $\diamondsuit$ restricts to a well-defined map $ \mathbf{Symm} \times \mathbf{Symm} \to \mathbf{Symm}$ (that is, the restriction of $\diamondsuit$ to $\mathbf{Symm} \times \mathbf{Symm}$ has its image in $\mathbf{Symm}$).

Is there any interesting combinatorics behind this map, just as the Gessel-Maia-Mendez species-theoretical interpretation of $\boxdot$? Does it take a nice form on some basis of $\mathbf{Symm}$ ?