  Let ${\mathcal O}$ be an involutive discrete valuation ring with residue
field of characteristic not 2. Let $A$ be a quotient of ${\mathcal O}$ by a
nonzero power of its maximal ideal and let $*$ be the involution that $A$
inherits from ${\mathcal O}$. We consider various unitary groups ${\mathcal
U}_m(A)$ of rank $m$ over $A$, depending on the nature of $*$ and the
equivalence type of the underlying hermitian or skew hermitian form. Each group
${\mathcal U}_m(A)$ gives rise to a Weil representation.
  In this paper, we give a Clifford theory description of all irreducible
components of the Weil representation of ${\mathcal U}_m(A)$ with respect to
all of its abelian congruence subgroups and a third of its nonabelian
congruence subgroups.
