"He sang everywhere, including in the bath".
"He behaved badly in many ways, including cheating in his exams."
These don't sound quite right to me. Are they wrong? And, if so, on the basis of what rule?
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Including as used here is a preposition. A participial preposition, but still a true preposition.
In English Grammar: A University Course by Angela Downing & Philip Locke is found this analysis on where prepositional phrases may be used:
Realisations of the complement element of a prepositional phrase
The complement element of a PP is most typically realised by a nominal group, but it may also be realised by the classes of groups and clauses shown below. Simple nouns and pronouns, adjectives and adverbs are treated as 'groups' represented by the head:
at home / after which / on account of his age
in private / at last / for good
for ever / since when / until quite recently
from out of the forest / except in here
(+ finite wh-clauses; wh- + to-infinitive clauses and ...
-ing clauses) [re-formatted]
They authors go on to mention restrictions that may apply with each type of element, but none are mentioned for prepositional phrases.
This 'licenses' "He sang everywhere, including in the bath". I'd say it's in an informal register.
With "He behaved badly in many ways, including cheating in his exams.", we have the ing-clause complement. Provided that the preceding main clause licenses the 'including', an ing-clause complement should be acceptable. With the second example here, one could paraphrase as "He behaved badly in many ways, one of which was cheating in his exams." I'd say this is another acceptable usage, though perhaps even more informal.