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We study the validity of the Blackadar-Kirchberg conjecture for extensions ofseparable, nuclear, quasidiagonal $C^*$-algebras that satisfy the UCT. Morespecifically, we show that the conjecture for the extension has an affirmativeanswer if the ideal lies in a class of $C^*$-algebras that is closed underlocal approximations and contains all separable ASH algebras, as well ascertain classes of simple, unital $C^*$-algebras and crossed products of unital$C^*$-algebras with the integers.

Extensions of quasidiagonal $C^*$-algebras and controlling the $K_0$-map of embeddings