  In [C. Ding, An infinite family of Steiner systems $S(2,4,2^m)$ from cyclic
codes, {\em J. Combin. Des.} 26 (2018), no.3, 126--144], Ding constructed a
family of Steiner systems $S(2,4,2^m)$ for all $m \equiv 2 \pmod{4}$ from a
family of extended cyclic codes. The objective of this paper is to present a
family of Steiner systems $S(2,4,2^m)$ for all $m \equiv 0 \pmod{4}$ supported
by a family of extended cyclic codes. The main result of this paper complements
the previous work of Ding, and the results in the two papers will show that
there exists a binary extended cyclic code that can support a Steiner system
$S(2,4,2^m)$ for all even $m \geq 4$. This paper also determines the parameters
of other $2$-designs supported by this family of extended cyclic codes.
