Black's priority is to organize his defence and to counter White's attack. Therefore, 5...Nxe4 is bad (moreover, White can win the knight with 6. Qh5+ g6 7. Qd5+).

There are three strategies to build comfortably the defence:

5...c5 tries to regain space by attacking the center from the queen side.

5...Be7 is a first step toward an "artificial castling". This prepares Re8 and Kg8.

5...g6 prepares a fianchetto. Black will leave the e file open and builds another kind of defence with the bishop on g7. But we may wonder if this will resist a White pawn advance on the king side.

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1. e4 e5 2. Nf3 Nf6 3. Nxe5 d6 4. Nxf7 Kxf7 5. d4

Wikipedia has related information at Petrov Defence.