Mean value property in Lie groups

Question

On $\mathbb{R}^n$, two equivalent definitions of harmonic functions are

1. $\Delta f=0$, where $\Delta$ is the Laplace operator.

2. $f$ satisfies the mean value property, i.e. $$ f(x)=\frac{1}{vol(B(x,r))}\int_{B(x,r)}fdV $$ for all $x\in \mathbb{R}^n$.

Does the same equivalency hold in Lie groups? that is, are there analogues definitions for Laplace operator, metric, and volume, for which $\Delta f=0$ is equivalent to mean value property?

Thanks!