In a manner that involves or relates to adultery or infidelity.
From 'adulterous' (adjective) plus the -ly adverbial suffix. This creates the adverbial form to describe how an action is done or how something is characterized.
The -ly suffix converting adjectives to adverbs is so ancient that it actually comes from Old English '-lice' (in a like manner), making 'adulterously' literally mean 'in an adulterer-like manner.'
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