  Let $k$ be an algebraically closed field of prime characteristic $p$. Let
$kGe$ be a block of a group algebra of a finite group $G$, with normal defect
group $P$ and abelian $p'$ inertial quotient $L$. Then we show that $kGe$ is a
matrix algebra over a quantised version of the group algebra of a semidirect
product of $P$ with a certain subgroup of $L$. To do this, we first examine the
associated graded algebra, using a Jennings--Quillen style theorem.
  As an example, we calculate the associated graded of the basic algebra of the
non-principal block in the case of a semidirect product of an extraspecial
$p$-group $P$ of exponent $p$ and order $p^3$ with a quaternion group of order
eight with the centre acting trivially. In the case $p=3$ we give explicit
generators and relations for the basic algebra as a quantised version of $kP$.
As a second example, we give explicit generators and relations in the case of a
group of shape $2^{1+4}:3^{1+2}$ in characteristic two.
