Pro-isomorphic zeta functions of nilpotent groups and Lie rings under base extension

We consider pro-isomorphic zeta functions of the groups $\Gamma (\mathcal {O}_K)$, where $\Gamma$ is a unipotent group scheme defined over $\mathbb {Z}$ and $K$ varies all number fields. Under certain conditions, we show that these have fine Euler decomposition with factors indexed by primes $\mathfrak {p}$ depending only on structure $\Gamma$, degree $[K:\mathbb {Q}]$, cardinality residue field $\mathcal {O}_K / \mathfrak {p}$. satisfy uniform rationality study their dependence {Q}]$. Explicit computations are given for several families groups.