Here's a one-relator ("large") example of a torsion free group such that the group of inner automorphisms has torsion:
$$\langle a_{1},a_{2},a_{3} : a_{1}^{2}=a_{2}^{3}, a_{2}^{5}=a_{3}^{7}\rangle$$.
This group is one-relator on $a_{1}$ and $a_{3}$, and by a result of Murasugi is torsion-free. The center of the group is generated by $a_{1}^{10}$.
This fact is mentioned in the first paragraph of this readily googlable paper of James McCool: A class of one-relator groups with centre, Bulletin of the Australian Mathematical Society, Volume 44, Issue 2, 2009.
The result is proved in this other readily googlable paper of Medskin, Pietrowski and Steinberg: One relator groups with center, Journal of the Australian Mathematical Society, Volume 16, 1973.
The result of Murasugi can be found in: The center of a group with a single defining relation, Math. Ann., 155, 1964.
