  We define a higher-order generalisation of the CPM construction based on
arbitrary finite abelian group symmetries of symmetric monoidal categories. We
show that our new construction is functorial, and that its closure under
iteration can be characterised by seeing the construction as an algebra for an
appropriate monad. We provide several examples of the construction, connecting
to previous work on the CPM construction and on categorical probabilistic
theories, as well as upcoming work on higher-order interference and
hyper-decoherence.
