Who will be responsible for council tax depends on what type of tenancy your tenant has and if the initial tenancy end date has passed.
A tenant with a tenancy for six months or more holds a material interest in the property and if they move out before the end of their tenancy, they are liable for unoccupied charges until their tenancy end date.
Once a tenant’s tenancy expires, if they stay on without a new tenancy being drawn up their tenancy becomes a statutory periodic or rolling tenancy.
This means they no longer hold a material interest in the property unless there is a continuation clause in the tenancy agreement.
So if a tenant’s six month tenancy has ended and has become a rolling or periodic tenancy, they usually cannot be held liable for any empty charges once they move out unless their tenancy agreement contains a continuation clause.
If there is a continuation clause that states the tenancy will continue after the expiry of the initial term, then the tenancy will continue, not come to an end. So no statutory periodic or rolling tenancy is created and the tenant will remain liable until the tenancy is terminated even if they vacate the property.
You should be careful not to accept an invalid notice or early end to a tenancy.
It could be seen as accepting an early end to the tenancy if the tenant is allowed to return the keys before the tenancy ends.