Permission for lodger in AST

I rented out a property to a couple and their cousin. The cousin is now leaving, and they want to bring in a new tenant. Should I change the tenancy agreement to be solely in the couple’s name and allow them to have a lodger? If so, can I simply grant permission for the lodger via email? Please advise.

It’s not a lodger unless it’s a live in landlord.
The new person is either a tenant or permitted occupier.

Write the contract for who’s paying rent. If the new person is not credit checked and the other two are then leave that person as permitted occupier if they are going to change the third person
Don’t take rent of said person though and no bills should be in his name .
You’ll need to start all over again serving new documents exit inventory check in inventory deposit etc if names change on contract

A House in multiple occupation (HMO) is a property rented out by at least 3 people who are not from 1 ‘household’ (for example a family) but share facilities like the bathroom and kitchen.

You need to be sure if you are creating a HMO, and if so, what extra fire precautions or Local authority licencing etc you will be governed by.

The new occupant would be a lodger, (licensee) if the couple continue to live there, but the exemption from HMO regs only applies to owner occupiers, so the property would be an HMO.