Client did not advise Housing Benefit and DWP she was leaving the UK

Client missed two HB payments to landlord- so he advised Local Authority who contacted client- She advised them she was overseas and had been for a couple of months. Local Authority have now suspended HB whilst it is reviewed- they asked client if she had informed DWP she had been out of country ( she had multiple trips totalling 26 weeks in last year) as she is in receipt of ESA and PIP- she advised L/A she did not inform them.
She has asked if she thinks local authority will inform DWP and if she she should pre-empt this and inform them herself.
She is also concerned if any further legal action may be taken against her other than having to repay any overpayment?

I am cautious to given any advise in this area but was wondering what others thoughts were.

Thanks