Question about permutation equivariance

[XikunHuang]

Thank you for the wonderful and inspiring work. I have a question about the permutation equivariance of GNN score function.
In the last paragraph of Section 3.2, it writes:

Note that since the message-passing operations of GNNs and the attention function used in GMH are permutation equivariant.

I am wondering what is the meaning of permutation equivariant here. To my understanding, a GNN is permutation equivariant means:
$$PH = GNN(PX, PAP^T)$$
where H is the node embedding.

However, for score function with respect to $A_t$, the output of $s(X_t, A_t, t)$ has the same dimensionality as $A_t$, but we treat the output as "node embedding" with dimension $n$, and the permutation equivariance means:
$$s(PX_t, PA_tP^T, t) = Ps(X_t, A_t, t)$$
not
$$s(PX_t, PA_tP^T, t) = Ps(X_t, A_t, t)P^T$$
right?

Looking forward to your reply. Thanks a lot.
Xikun