When I say serious crime, maybe I should have said, anything that can’t be dealt with by The CofC using normal military rank structure, insubordination, dress offences, failing to carry out a duty etc. A crime that requires a criminal investigation.
Why would HO forces investigate purely service crime, other than where it is a statutory requirement in UK, the main one being murder? They have enough work to do, the SP police are funded and remitted to do that, along with The MOD Police. I think the RNP, are a slight outlier in The SP, as their ship role is probably more akin to a Regimental Police NCO, in that they are responsible for unit discipline. Not something RMP would necessary get involved in.