Opener's rebid of 2NT (system: Simple SAYC)
The SAYC card specifies strong jump shifts by responder after a 1H opener, further explaining that it invites a slam. This suggests a 17-19 HCP range with 5+ cards and a good honor holding in the bid suit (though I warn you that point-count is insufficient to evaluate many slam invitational hands, and e.g. AKJxxxx Axx Kx x is a 15-count that suggests a strong jump shift).

Though 2NT only promises a minimum opener as explained by Aryabhata, your first priority must be to show your support for partner's spade suit. Your proper rebid is 4S by the principle of fast arrival. You know enough to sign off: your hand is not good enough to accept a slam invitation (thus the game), and you know your partnership has a spade fit (thus the strain). Bidding 3S here suggests to partner that slam is still a possibility.

If you'd like to read more about opener's rebids after a strong jump shift by responder, the bridge bum has a good summary.

Edit The question has been updated to make the auction start 1H-2C instead of 1H-2S.

SAYC suggests that 1H-2C-2NT should show a balanced hand, 12-14 HCP. Be aware that if the club bidder is an unpassed hand, 1H-2C promises a second bid from responder, and so you're forcing your side to the three level with a 2NT response. You should probably consider a different rebid if your hand doesn't stop the other suits (especially spades, the other major, as defenders prefer to lead majors against no trump contracts).

1H-2C-2H does not promise a 6th heart (as opposed to 1H-1NT-2H or 1H-1S-2H, which do). So if your 5-3-3-2 12-count is something like Jxx AKJxx Qxx Jx, I would rebid 2H. If instead it's AJT KJxxx QJx xx, 2NT seems better.