Taylor series of function on Lie group and integral equation

Question

Given a smooth function Y from a matrix Lie group $G$ to $\mathbb{C}$ I want to calculate the Taylor expansion of $Y(Me^X)$ up to second order where $X$ is an element of the Lie algebra. Is it true that this yields \begin{equation} Y(Me^X)\approx{Y}(M)+(L_XY)(M)+(L_X^2Y)(M) \end{equation} where $L_X$ is the left invariant vector field associated with $X$? The origin of the problem is the following matrix integral equation \begin{equation} Y_1(M)=\int{DX}e^{-\frac{1}{t}\text{Tr}X^2}Y(Me^X) \end{equation} where the integral is over the Lie algebra of $G$ and the measure fulfills \begin{equation} \int{DX}e^{-\frac{1}{t}\text{Tr}X^2}=1 \end{equation} where $t>0$ is an auxiliary parameter. I want to compute the integral up to second order in the Taylor expansion of $Y$. If the above Taylor expansion is right I don't know how to proceed from there. It is clear that the zeroth order reproduces $Y(M)$ but I am not sure how the differential operators of the first and second order act on $Y$ and how to compute the matrix integral. My assumption is that one will get a reproduction of $Y(M)$ and a term containing the Laplace Beltrami operator but I am not sure how to prove this. I would be grateful for a little hint.