Fix a cardinal $\lambda$$\newcommand{\cU}{\mathcal U}\newcommand{\cV}{\mathcal V}$. Consider the equivalence relation on $\beta\lambda$ given by $\cU\sim \cV$ when for all first-order structures $M$ we have $M^{\cU}\cong M^{\cV}$.

By considering the structure $(\lambda,A)_{A\subseteq \lambda}$, one can show that the $\sim$-classes have at most $2^\lambda$ elements (arguing as here).

If we consider the action of the symmetric group $S(\lambda)$ on $\beta\lambda$, it is easy to see that for each $\cU\in \beta\lambda$ and $f\in S(\lambda)$, we have $\cU\sim f(\cU)$.

Is the converse also true, namely, if $\cU\sim \cV$, are they necessarily conjugate by a permutation of $\lambda$? If not, is it true for $\lambda=\aleph_0$ (or any $\lambda$)?

Fix a cardinal $\lambda$$\newcommand{\cU}{\mathcal U}\newcommand{\cV}{\mathcal V}$. Consider the equivalence relation on $\beta\lambda$ given by $\cU\sim \cV$ when for all first-order structures $M$ we have $M^{\cU}\cong M^{\cV}$.

By considering the structure $(\lambda,A)_{A\subseteq \lambda}$, one can show that the $\sim$-classes have at most $2^\lambda$ elements (arguing as here).

If we consider the action of the symmetric group $S(\lambda)$ on $\beta\lambda$, it is easy to see that for each $\cU\in \beta\lambda$ and $f\in S(\lambda)$, we have $\cU\sim f(\cU)$.

Is the converse also true, namely, if $\cU\sim \cV$, are they necessarily conjugate by a permutation of $\lambda$? If not, is it true for $\lambda=\aleph_0$ (or any $\lambda$)?

Edit: Given that the answer is positive (for signature of size $2^\lambda$), it would also be interesting to know whether the equivalence still holds if we define $\cU\sim \cV$ when $M^{\cU}\cong M^{\cV}$ for all first-order structures $M$ whose signature is either

1. strictly smaller than $\lambda$, or
2. of size at most $\lambda$, or
3. strictly smaller than $2^{\lambda}$.