Block components of generalized quaternion group codes

Codes in the generalized quaternion group algebra ${\mathbb{F}}_q[Q_{4n}]$ are considered. Restricting to char${\mathbb{F}}_q\nmid 4n$ the structure of an arbitrary code $C\subseteq {\mathbb{F}}_q[Q_{4n}]$ is described via the Wedderburn decomposition. Moreover it is known that in this case every code $C\subseteq {\mathbb{F}}_q[Q_{4n}]$ has a generating idempotent $\lambda \in {\mathbb{F}}_q[Q_{4n}]$. Given the generating idempotent of a code $C$ we determine the different components in its decomposition $C\cong {⨁}_{j=1}^{r+s}{C}_j\oplus {⨁}_{i=1}^{k+t}{C}_i^{\mathrm{\prime }}\mathrm{.}$ Afterwards we apply this result to describe the blocks of codes induced by cyclic group codes.