  Let $k$ be an algebraically closed field of characteristic 2, and let $G$ be
a finite group. Suppose $B$ is a block of $kG$ with dihedral defect groups such
that there are precisely two isomorphism classes of simple $B$-modules. The
description by Erdmann of the quiver and relations of the basic algebra of $B$
is usually only given up to a certain parameter $c$ which is either 0 or 1. In
this article, we show that $c=0$ if there exists a central extension $\hat{G}$
of $G$ by a group of order 2 together with a block $\hat{B}$ of $k\hat{G}$ with
generalized quaternion defect groups such that $B$ is contained in the image of
$\hat{B}$ under the natural surjection from $k\hat{G}$ onto $kG$. As a special
case, we obtain that $c=0$ if $G=\mathrm{PGL}_2(\mathbb{F}_q)$ for some odd
prime power $q$ and $B$ is the principal block of $k
\mathrm{PGL}_2(\mathbb{F}_q)$.
