On algebraic and arithmetic properties of monoids of product-$K$ sequences

Let $G$ be a group and $K$ be a normal subgroup of $G$. A sequence over $G$ is a finite collection of terms from $G$, where repetition is allowed, and the order is disregarded. A product-$K$ sequence is a sequence whose terms can be ordered such that their product in $G$ belongs to $K$. The set $\ma