The theorem of Kaloujnine-Krasner says
Given two groups $D$ and $Q$, the wreath product $D \wr Q$ contains an isomorphic copy of every extension of $D$ by $Q$.
I am looking for an English reference of a complete proof.
In Rotman's Group Theory book, the theorem can be found for $Q$ finite (Theorem 7.37, fourth ed.) but this is not enough for my purposes. Supposedly, a full proof can be found in Kargapolov & Merzliakov's "Eléments de la théorie des groupes" but it's in French.
Can anyone point me to a proof written in English?
Try C. Wells, "Some applications of the wreath product construction", Amer. Math. Monthly 83 (1976), no. 5, 317–338.
While I cannot say whether your particular application occurs in that paper, I've found it to be a good source on wreath product stuff more generally, including some work of Kaloujnin-Krasner, so yours might be in there.