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Income support is assessed by the Department of Work and Pensions and any alleged fraud is investigated by their Audit and Investigations department. Any decision they make in relation to a claim for IS is relevant only to DWP benefits.
Housing Benefit is assessed by the Local Authority (e.g. Borough Council) who have nothing formally to do with the DWP. They may sometimes be informed by DWP if DWP believe a fraudulent claim has been made, for example if a bank account with a large amount of money in it which would invalidate the claim was not mentioned when IS was claimed then DWP may alert the Local Authority to allow them to check their records to see if the same account is an issue for them.
However, the simple answer is that DWP and LA independently assess each claim. Having an IS claim suspended does not automatically eliminate someone from claiming Housing Benefit.
I hope that assists but please feel free to ask follow questions.