On $\kappa$-homogeneous, but not $\kappa$-transitive permutation groups

A permutation group $G$ on a set $A$ is $\kappa$-homogeneous iff for all $X,Y\in [A{]}^{\kappa }$ with $\mathrm{\mid }A\setminus X\mathrm{\mid }=\mathrm{\mid }A\setminus Y\mathrm{\mid }=\mathrm{\mid }A\mathrm{\mid }$ there is a $g\in G$with$g[X]=Y$\mathrm{.}$G$is${\kappa}$-transitiveiffforanyinjectivefunction$f$with$dom(f)\cup ran(f)\in [A]^{\le {\kappa}}$and$|A\setminus dom(f)|=|A\setminus ran(f)|=|A|$thereisa$g\in G$with$f\subset g$. Giving a partial answer to a question of P. M. Neumann we show that there is an $\omega$-homogeneous but not $\omega$-transitive permutation group on a cardinal $\lambda$ provided (i) $\lambda <{\omega }_{\omega }$, or (ii) $2^{\omega }<\lambda$, and ${\mu }^{\omega }={\mu }^{+}$ and ${\mathrm{\square }}_{\mu }$ hold for each $\mu \le \lambda$ with $\omega =cf(\mu )<\mu$, or (iii) our model was obtained by adding ${\omega }_1$ many Cohen generic reals to some ground model. For $\kappa >\omega$ we give a method to construct large $\kappa$-homogeneous, but not $\kappa$-transitive permutation groups. Using this method we show that there exists ${\kappa }^{+}$-homogeneous, but not ${\kappa }^{+}$-transitive permutation groups on ${\kappa }^{+n}$ for each infinite cardinal $\kappa$ and natural number $n\ge 1$ provided $V=L$.