Parametrization and group operation

Let $A$ be a group of measurable transformation of $\mathcal{X} \subset \mathbb{R}^m$ into itself, such that each $a_\theta \in A$ is parameterized by $\theta \in \Theta \subset \mathbb{R}^d$.

I have difficulty understanding the second and third points of the following definition

1. Given any pair of points $\theta_0$ and $\theta_1$, let $\theta_{01}$ be such that $a_{\theta_{01}}=a_{\theta_{0}}^{-1} \circ a_{\theta_{1}}$. There exist open neighborhoods $V_0$ and $V_1$ and $V_{01}$ of $\theta_0$, $\theta_1$ and $\theta_{01}$ respectively such that, $\theta' \in V_0$ and $\theta \in V_1$, implies $a_{\theta'}^{-1} \circ a_{\theta}=a_{\theta^*}$ for some $\theta^* \in V_{01}$.

2. The mapping $\theta\rightarrow a_\theta$ is one-to-one on $ V_{01}$. Let $\nu$ denote its local inverse on the image of $V_{01}$.

3. Define the mapping from $V_0 \times V_1 \rightarrow V_{01}$ by\begin{equation}\lambda(\theta', \theta) = \nu(a_{\theta'}^{-1} \circ a_{\theta}).\end{equation}Then the mapping $\omega(\theta)=\lambda(\theta_0, \theta)$ has derivative $\dot{\omega}(\theta_0)$ at $\theta_0$ (that is an $k\times k$-matrix).

I cannot understand what is the local inverse $\nu$ in Point 2 and then of Point 3.

For example, let's take the parameterized affine transformation $a_\theta(x) = p(\theta) + S(\theta)x$, where $p(\theta)$ is a continuous $m$-dim vector valued function and $S(\theta)$ is a continuous $m\times m$-matrix valued function.

How can I explicitly express $\omega(\theta)$ and $\dot{\omega}(\theta_0)$ for this example?