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We investigate univalent functions $f(z)=z+a_2z^2+a_3z^3+\ldots$ in the unitdisk $\mathbb D$ extendible to $k$-q.c.(=quasiconformal) automorphisms of$\mathbb C$. In particular, we answer a question on estimation of $|a_3|$raised by K\"uhnau and Niske [Math. Nachr. 78 (1977) 185-192]. This is one ofthe results we obtain studying univalent functions that admit q.c.-extensionsvia a construction, based on Loewner's parametric representation method, due toBecker [J. Reine Angew. Math. 255 (1972) 23-43]. Another problem we consider isto find the maximal $k_*\in(0,1]$ such that every univalent function $f$ in$\mathbb D$ having a $k$-q.c. extension to $\mathbb C$ with $k\leqslant k_*$admits also a Becker q.c.-extension, possibly with a larger upper bound for thedilatation. 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$k\in(0,1)$, to each finite Blaschke product there corresponds a univalentfunction $f$ in $\mathbb D$ that admits a Becker $k$-q.c. extension but no$k'$-q.c. extensions to $\mathbb C$ with $k'<k$.

Univalent functions with quasiconformal extensions: Becker's class and estimates of the third coefficient