We investigate univalent functions f (z) = z + a 2 z 2 + a 3 z 3 + . . . in the unit disk D extendible to k-q.c.(=quasiconformal) automorphisms of C. In particular, we answer a question on estimation of |a 3 | raised by Kühnau and Niske [Math. Nachr. 78 (1977) 185-192]. This is one of the results we obtain studying univalent functions that admit q.c.-extensions via a construction, based on Loewner's parametric representation method, due to Becker [J. Reine Angew. Math. 255 (1972) 23-43]. Another problem we consider is to find the maximal k * ∈ (0, 1] such that every univalent function f in D having a k-q.c. extension to C with k k * admits also a Becker q.c.-extension, possibly with a larger upper bound for the dilatation. We prove that k * > 1/6. Moreover, we show that in some cases, Becker's extension turns out to be the optimal one. Namely, given any k ∈ (0, 1), to each finite Blaschke product there corresponds a univalent function f in D that admits a Becker k-q.c. extension but no k -q.c. extensions to C with k < k.