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Use a rabbeting bit and a plunge router. (A Forstner bit can't be used to enlarge a hole -- there’s no center to orient the bit properly so that the hole ends up the right size and in the right place.) A rabbeting bit has a bearing in the center that rides along the edges of the inside of the hole to help guide the bit (Image 1).
Plunge into the hole as far as the bit will go. The bit may not be long enough to cut all the way through, but it is a relatively large bit, so run the router at a slower speed while cutting into the wood (Image 2).