Let $G$ be a finite group and $C_2$ the cyclic group of order $2$. Consider the $8$ multiplicative operations $(x,y)\mapsto {\left(x^i{y}^j\right)}^k$, where $i$, $j$, $k\in \{-1,1\}$. Define a new multiplication on $G\times C_2$ by assigning one of the above $8$ multiplications to each quarter $(G\times \{i\left\}\right)\times (G\times \{j\left\}\right)$, for $i,j\in C_2$. We describe all situations in which the resulting quasigroup is a Bol loop. This paper also corrects an error in P. Vojtěchovsk’y: On the uniqueness of loops $M(G,2)$.

A loop $Q$ is said to be left conjugacy closed (LCC) if the set $\{L_x;x\in Q\}$ is closed under conjugation. Let $Q$ be such a loop, let $ℒ$ and $ℛ$ be the left and right multiplication groups of $Q$, respectively, and let $InnQ$ be its inner mapping group. Then there exists a homomorphism $ℒ\to InnQ$ determined by $L_x\mapsto R_x^{-1}{L}_x$, and the orbits of $[ℒ,ℛ]$ coincide with the cosets of $A\left(Q\right)$, the associator subloop of $Q$. All LCC loops of prime order are abelian groups.

Let $Q$ be a finite commutative loop and let the inner mapping group $I\left(Q\right)\cong C_{p^n}\times C_{p^n}$, where $p$ is an odd prime number and $n\ge 1$. We show that $Q$ is centrally nilpotent of class two.

Let $G$ be a finite group and $C_2$ the cyclic group of order 2. Consider the 8 multiplicative operations $(x,y)\mapsto {\left(x^i{y}^j\right)}^k$, where $i,j,k\in \{-1,1\}$. Define a new multiplication on $G\times C_2$ by assigning one of the above 8 multiplications to each quarter $(G\times \{i\left\}\right)\times (G\times \{j\left\}\right)$, for $i,j\in C_2$. If the resulting quasigroup is a Bol loop, it is Moufang. When $G$ is nonabelian then exactly four assignments yield Moufang loops that are not associative; all (anti)isomorphic, known as loops $M(G,2)$.