  We prove the subnormality of an operator-weighted composition operator whose
symbol is a transformation of a discrete measure space and weights are
multiplication operators in $L^2$-spaces under the assumption of existence of a
family of probability measures whose Radon-Nikodym derivatives behave regular
along the trajectories of the symbol. We build the quasinormal extension which
is a weighted composition operator induced by the same symbol. We give
auxiliary results concerning commutativity of operator-weighted composition
operators with multiplication operators.
