Let $E$ be path-connected. Let $p : E → B$ be a covering map and $p_∗$ be the induced homomorphism from the fundamental group of $E$ to the fundamental group of $B$. Let $e_o$ and $e_1$ be points in E such that $p(e_0) = p(e_1) = b$. Consider the subgroups $G_0 = p_∗π_1(E, e_0)$ and $G_1 = p_∗π_1(E, e_1)$.
How can I prove that there exists $γ ∈ π_1(B, b)$ such that every $σ ∈ G_0$ can be written uniquely in the form $γτ γ^{-1}$ for some $τ ∈ G_1$?
I like to do such situations in the algebraic setting of covering morphisms of groupoids, see Chapter $10$ of the book Topology and Groupoids. So if $p:E \to B$ is a covering map of spaces then $\pi_1 p: \pi_1 E \to \pi_1 B$ is a covering morphism of groupoids. So $q: H \to G$ is a covering morphism of groupoids if for each $e \in Ob(H)$ and each $g: p(e) \to b$ in $G$ there is a unique $h: e \to e'$ in $H$ such that $q(h)=g$. This easily implies that a composable sequence $g_1, \ldots,g_n$ starting at $b$ has a unique lift to a composable sequence $(h_1,\ldots,h_n) $ starting at $e$ of elements of $H$ such that $q(h_i)=g_i, i=1, \ldots, n$.
Now suppose $H$ is connected and $q(e_0)=q(e_1)=b$. Since $H$ is connected there is an element $h: e_0 \to e_1$ of $H$. Let $\gamma=q(h): b \to b$. Let $G_i= q(H(e_i)), i=0$, where $H(e)$ is the group of elements of $H$ which are $e \to e$. We know by unique lifting that $q$ is injective on $H(e)$.
Let $\sigma \in G_0$. You should now be able to use unique lifting to prove the result you want. Draw a picture of the situation in $H$.