Let $A$ be an annulus, and consider ${\rm Mod}(A, \{x,y\})$, the group of connected components of homeomorphisms of $A$ that fix its two boundary components pointwise, and preserve the set $\{x, y\}$. I would like to explicitly understand what this group is, perhaps with a presentation.
I can see there is a homomorphism $\varphi\colon{\rm Mod}(A, \{x, y\})\to{\rm Mod}(D^2, \{x,y,z\})\cong B_3=\langle\sigma_1,\sigma_2 | \sigma_1\sigma_2\sigma_1=\sigma_2\sigma_1\sigma_2\rangle$ the braid group on three strands, given by capping off one of the boundary components of $A$ with disk with one marked point, which turns $A$ into a disk $D^2$ with three marked points.
Its image is the set of braids that preserve the new third point $z$. This subgroup of $B_3$ is $\langle a:=\sigma_1, b:=\sigma_2^2\rangle$, and in fact has a presentation: $\langle a,b \mid abab=baba\rangle$. This group is non-abelian, so I know the mapping class group I'm interested in is non-abelian.
I think I am looking for a certain central extension of this subgroup of $B_3$. But how do I know which one? I'm not sure.