I am reading a research paper in which group $G=D_{30}=\{x,y\mid x^{15}=y^2=1, yxy=x^{-1}\}$ and $K$ be the normal subgroup of $G$ generated by $x^5$. Then $G/K\cong H\cong \langle x^3,y\rangle.$ Thus from the the ring epimorphism $$FG\to FH$$ given by $$\sum_{j=0}^4\sum_{i=0}^2x^{5i+3j}(a_{i+3j}+a_{i+3j+15}y\to \sum_{j=0}^4\sum_{i=0}^2x^{3j}(a_{i+3j}+a_{i+3j+15}y$$ we get a ring monomorphism $$FH\to FG$$ $$\sum_{i=0}^4x^{3i}(b_i+b_{i+5}y)\to\sum_{i=0}^4x^{3i}(b_i+b_{i+5}y)$$ Throughout $F$ is a finite field of characteristic $3$ and $FG$ is the group algebra.
My question is I don’t know how these epimorphism and monomorphism are and how the authors think about it. Please help. Thank you in advance. http://www.ieja.net/files/papers/volume-29/10-V29-2021.pdf