Grutch is an somewhat outdated word, perhaps at the point of being obsolete. It means to grumble, to complain, to find fault with, or to be angry at something. Old English stole grouchier or grocier from old French, and 'grutch' was the anglicized outcome.
By the 1400s it came to mean 'to envy someone for possessing something'; by the late 1500s it evolved into grudge, and by the late 1500s it added a be- to give us the modern begrudge.
Grutch also evolved to give us gruccild, meaning 'a woman who complains' (now obsolete) and grouch; meanwhile grudge also spawned the adjectives grudgingly and begrudgingly.
Grutch has also made appearances as a noun meaning a complaint, malice, or ill-will; as this usage appeared in the once-famous poem Hudibras, by Samuel Butler, this definition appears in many older dictionaries, but it is unclear how common it was in spoken usage.