What's that process called where you form new group law by $x \star y = x \cdot a \cdot y$ for some $a$ in the group? Let $A$ be an abelian group.  We can form new groups $(A, \cdot a \cdot)$ where $a$ is any element of $A$.  Choosing $a = 1$ the identity of $A$ gives $A$ itself.
Clearly, associativity comes from associativity and commutativity of $A$.
Identity is $x \cdot a \cdot e = x$ or $e = a^{-1}$.
Inverse is $x \cdot a \cdot y = a^{-1}$ or $y = a^{-1} x^{-1} a^{-1}$.
What is this group formation process called and do the groups relate back to $A$?
I've seen it somewhere, and can't find where again.

I have seen this also for groups, but it seems to be more interesting for Lie algebras, see here and here. For groups it seems to be called variant, see the comments at the above questions, or a sandwich, e.g., Semigroups under a sandwich operation, Proc. Edinburgh Math. Soc. (Ser. 2) 26 (1983), 371-382.
For $K$-algebras $(A,\cdot)$ with multiplication $x\cdot y$ a new multiplication
$x\circ_z y$ depending on a fixed $z$ is called a homotope, or a mutation. 
