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Groups with commuting inner mappings are of nilpotency class at most two, butthere exist loops with commuting inner mappings and of nilpotency class higherthan two, called loops of Cs\"org\H{o} type. In order to obtain small loops ofCs\"org\H{o} type, we expand our programme from `Explicit constructions ofloops with commuting inner mappings', European J. Combin. 29 (2008), 1662-1681,and analyze the following setup in groups:  Let $G$ be a group, $Z\le Z(G)$, and suppose that $\delta:G/Z\times G/Z\to Z$satisfies $\delta(x,x)=1$, $\delta(x,y)=\delta(y,x)^{-1}$,$z^{yx}\delta([z,y],x) = z^{xy}\delta([z,x],y)$ for every $x$, $y$, $z\in G$,and $\delta(xy,z) = \delta(x,z)\delta(y,z)$ whenever $\{x,y,z\}\cap G'$ is notempty.  Then there is $\mu:G/Z\times G/Z\to Z$ with $\delta(x,y) =\mu(x,y)\mu(y,x)^{-1}$ such that the multiplication $x*y=xy\mu(x,y)$ defines aloop with commuting inner mappings, and this loop is of Cs\"org\H{o} type (ofnilpotency class three) if and only if $g(x,y,z) =\delta([x,y],z)\delta([y,z],x)\delta([z,x],y)$ is nontrivial.  Moreover, $G$ has nilpotency class at most three, and if $g$ is nontrivialthen $|G|\ge 128$, $|G|$ is even, and $g$ induces a trilinear alternating form.We describe all nontrivial setups $(G,Z,\delta)$ with $|G|=128$. This allows usto construct for the first time a loop of Cs\"org\H{o} type with an innermapping group that is not elementary abelian.

Small loops of nilpotency class three with commutative inner mapping groups