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Self-dual codes over finite fields and over some finite rings have been ofinterest and extensively studied due to their nice algebraic structures andwide applications. Recently, characterization and enumeration of Euclideanself-dual linear codes over thering~$\mathbb{F}_{q}+u\mathbb{F}_{q}+u^2\mathbb{F}_{q}$ with $u^3=0$ have beenestablished. In this paper, Hermitian self-dual linear codes over$\mathbb{F}_{q}+u\mathbb{F}_{q}+u^2\mathbb{F}_{q}$ are studied for all squareprime powers~$q$. Complete characterization and enumeration of such codes aregiven. Subsequently, algebraic characterization of $H$-quasi-abelian codes in$\mathbb{F}_q[G]$ is studied, where $H\leq G$ are finite abelian groups and$\mathbb{F}_q[H]$ is a principal ideal group algebra. General characterizationand enumeration of $H$-quasi-abelian codes and self-dual $H$-quasi-abeliancodes in $\mathbb{F}_q[G]$ are given. For the special case where the fieldcharacteristic is $3$, an explicit formula for the number of self-dual $A\times\mathbb{Z}_3$-quasi-abelian codes in $\mathbb{F}_{3^m}[A\times\mathbb{Z}_3\times B]$ is determined for all finite abelian groups $A$ and $B$such that $3\nmid |A|$ as well as their construction. Precisely, such codes canbe represented in terms of linear codes and self-dual linear codes over$\mathbb{F}_{3^m}+u\mathbb{F}_{3^m}+u^2\mathbb{F}_{3^m}$.

Self-Dual Linear Codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}+u^2\mathbb{F}_{q}$ and Their Applications in the Study of Quasi-Abelian Codes