Let $K$ be a CM-field with totally real subfield $F$. Let $(V_1, h_1)$ and $(V_2, h_2)$ be two $n$-dimensional $K$-vector spaces with nondegenerate Hermitian forms, where $n\geq 3$. 

> **Question 1** Does every $F$-group isomorphism $PU(V_1, h_1)\cong PU(V_2, h_2)$ arise from an isometry of $(V_1, h_1)\cong (V_2, \lambda h_2)$  for some $\lambda\in F^{\times}$ ? 

> **Edit. Question 2** Suppose there exists an $F$-group isomorphism $PU(V_1, h_1)\cong PU(V_2, h_2)$, can we deduce $(V_1, h_1)\cong (V_2, \lambda h_2)$ for some $\lambda\in F^{\times}$, or any other relations between $h_1$ and $h_2$?

A related statement for orthogonal group is given in the comment to the question https://mathoverflow.net/questions/226214/p-adic-orthogonal-groups-in-four-variables .

We are interested in the case that $K$ is a cyclotomic field and $h$ has signature $(1,n-1)$ for an embedding $F\to \mathbb{R}$ and is definite for all the other embeddings.