  We construct consistent Kaluza-Klein truncations of type IIA supergravity on
(i) $\Sigma_2\times S^3$ and (ii) $\Sigma_3\times S^3$, where $\Sigma_2 =
S^2/\Gamma$, $\mathbb{R}^2/\Gamma$, or $\mathbb{H}^2/\Gamma$, and $\Sigma_3 =
S^3/\Gamma$, $\mathbb{R}^3/\Gamma$, or $\mathbb{H}^3/\Gamma$, with $\Gamma$ a
discrete group of symmetries, corresponding to NS5-branes wrapped on $\Sigma_2$
and $\Sigma_3$. The resulting theories are a $D=5$, $\mathcal{N}=4$ gauged
supergravity coupled to three vector multiplets with scalar manifold
$SO(1,1)\times SO(5,3)/(SO(5)\times SO(3))$ and gauge group
$SO(2)\times\left(SO(2)\ltimes_{\Sigma_2}\mathbb{R}^4\right)$ which depends on
the curvature of $\Sigma_2$, and a $D=4$, $\mathcal{N}=2$ gauged supergravity
coupled to one vector multiplet and two hypermultiplets with scalar manifold
$SU(1,1)/U(1)\times G_{2(2)}/SO(4)$ and gauge group
$\mathbb{R}^+\times\mathbb{R}^+$ for truncations (i) and (ii) respectively.
Instead of carrying out the truncations at the 10-dimensional level, we show
that they can be obtained directly by performing In\"on\"u-Wigner contractions
on the 5 and 4-dimensional gauged supergravity theories that come from
consistent truncations of 11-dimensional supergravity associated with M5-branes
wrapping $\Sigma_2$ and $\Sigma_3$. This suggests the existence of a broader
class of lower-dimensional gauged supergravity theories related by group
contractions that have a 10 or 11-dimensional origin.
