Pluperfect ἐβεβήκη in Reading Greek

I am doing the exercises in section 16A-B of Reading Greek. We are to turn the perfects into pluperfects. For βέβηκα, I would have written ἐβέβηκει, but the study guide gives the form in my title. I believe that ending is attested in the literature, but perhaps not for this verb. Is this a mistake in the guide? As always, I would be grateful for any comments.

Are we talking about the 1s, or the 3s?

For the 3s, (ἐ)βεβήκει occurs 34 times in the Homer (usually without the augment).

Since the perfect form they provide is 1s, I would assume that they want you to make that into the pluperfect 1s, which would be -η.

(Smyth sec 455 has a note that -κεα can be contracted to -κη in the pluperfect, but if I’m understanding correctly, that wouldn’t seem to apply to the 3s of this verb. I think there are some cases where a -ε- can be added to the stem, Smyth 485. But AFAIK that doesn’t happen for this verb.)

Thank you very much. I see my error.