Let $G$ and $H$ be Lie groups and let $\phi\colon G\to H$ be a Lie group homomorphism. If $X\in\mathfrak{X}(G)$ is left invariant, is $\phi_*X\in\mathfrak{X}(H)$ (the pushforward of $X$ under $\phi$) left invariant, too?
Here is why I ask this: Let $e$ and $e'$ be the neutral elements of $G$ and $H$. In the book I am currently reading, $\mathfrak{g}\subset\mathfrak{X}(G)$ is the set of left invariant vector fields and $\phi_*\colon\mathfrak{g}\to\mathfrak{h}$ is defined by \begin{equation} (\phi_*X)_{e'}=\mathrm{d}\phi_eX_e, \end{equation} where the right side is just the usual pushforward evaluated at $e=\phi^{-1}(e')$ and one needs to remember that left-invariant vector field are uniquely determined by their value at one point.
Of course, that leads to the question whether two different vector fields are denoted by $\phi_*X$, the usual pushforward and the left invariant vector field with the same value at $e'$. If the answer to my question is yes, this is not a problem, since $\phi_*\colon\mathfrak{g}\to\mathfrak{h}$ is just the restriction of the pushforward $\phi_*\colon\mathfrak{X}(G)\to\mathfrak{X}(H)$ to left invariant vector fields.