Abstract
It is known that if $ G=AB $ is a product of its totally permutable subgroups $ A $ and $ B $, then $ Gin mathfrak{F} $ if and only if $ Ain mathfrak{F} $ and $ Bin mathfrak{F} $ when $ mathfrak{F} $ is a Fischer class containing the class $ mathfrak{U} $ of supersoluble groups. We show that this holds when $ G=AB $ is a weakly totally permutable product for a particular Fischer class, $ mathfrak{F}diamond mathfrak{N} $, where $ mathfrak{F} $ is a Fitting class containing the class $ mathfrak{U} $ and $ mathfrak{N} $ a class of nilpotent groups. We also extend some results concerning the $ mathfrak{U} $-hypercentre of a totally permutable product to a weakly totally permutable product.
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