Extending a principal connection form to an associated principal bundle Suppose that I have a principal $G$-bundle $\mathcal{G} \to M$ equipped with a principal connection form $\omega_\mathcal{G}: T\mathcal{G} \to \mathfrak{g}$ and a lie group homorphism $\varphi: G \to H$. Explicitly, what is the formula for the extended principal connection form on the associated principal $H$-bundle
$\mathcal{G} \times_G H$?  I've tried looking in various textbooks but haven't found this discussed.
This should be a linear map $T \mathcal{G} \times_{TG} TH \cong T(\mathcal{G} \times_G H)  \to \mathfrak{g}$. My guess is that it would be obtained from the sum of the principal connection form $\omega_\mathcal{G}$ and the Maurer-Cartan form $\omega_H$ on elements in $T \mathcal{G} \times  TH$. But I would need to show that this is a $TG$-balanced map so that it descends to the quotient $T \mathcal{G} \times_{TG} TH$.
I can show that the Maurer-Cartan form transforms with respect to the tangent group structure as
$$\omega_H((\varphi_* X) \cdot Y) = \operatorname{Ad}(h)\varphi_*\omega_G(X) + \omega_H(Y),$$
for $X \in T_g G$ and $Y \in T_h H$
But it seems more difficult to figure out how the principal connection form should behave with respect to the action of the tangent group $TG$ since we lack an explicit formula for the connection form.
 A: Before discussing connection forms, let's discuss connections. Consider G-bundle $\mathcal{G}\to M$. At every point $u \in \mathcal{G}$ there is a vertical subspace $G_u \subset T_u\mathcal{G} = \{A^*_u \colon A \in \mathfrak{g}\}$, where $A^*_u = L_{u*}A$. Assume we have a connection. That is, for every $u$ we have $Q_u \subset T_u\mathcal{G}$ s.t. we have a direct sum $T_u\mathcal{G} = G_u + Q_u$, and $Q_{ua} = (R_a)_* Q_u$ for every $a\in G$.
Given Lie group homomorhpism $\varphi: G\to H$ we have the associated map
$$\widetilde\varphi_{b} \colon \mathcal{G} \to \mathcal{G} \times_G H: u \mapsto \overline{(u,b)}\text{ for any }b\in H. \tag{1}$$
This map induces the corresponding map of tangent spaces:
$$ \widetilde\varphi_{b*} \colon T_u\mathcal{G} \to T_{\widetilde{\varphi}_b(u)}(\mathcal{G} \times_G H). \tag{2}$$
Therefore, we get $T_{\widetilde{\varphi}_b(u)}(\mathcal{G} \times_G H) = H_{\widetilde{\varphi}_b(u)} + \widetilde{\varphi}_{b*}(Q_{u})$, where $H_v$ for $v \in \mathcal{G} \times_G H$ is
the vertical subspace defined similarly to $G_u$. One can check that $\widetilde \varphi_{bc} = R_c \circ \widetilde\varphi_{b}$ and $\widetilde\varphi_{\varphi(a)b} = \widetilde\varphi_b \circ R_a$ for $a\in G, b,c\in H$ and, consequently, $Q_{Hv} = \widetilde{\varphi}_{b*}(Q_{u})$ for $\widetilde \varphi_b(u) = v$ is well defined and invariant with respect to $R_b$.
Now consider connection forms. Let $\omega_{\mathcal{G}}$ be the connection form on $\mathcal{G}$ and $\omega_{\mathcal{G}\times_G H}$ be the connection form on $\mathcal{G}\times_G H$. In order to compute $\omega_{\mathcal{G}}(X_u)$ we could
decompose $X_u = A^*_u + Y_u$, where $Y_u \in Q_u$, and write $\omega_{\mathcal{G}}(X_u) = A \in \mathfrak{g}$.
Similarly, for $\omega_{\mathcal{G}\times_G H}$, we decompose $X_v = B^*_v + Y_v$, where $Y_v \in Q_{Hv}$, and write $\omega_{\mathcal{G}\times_G H}(X_v) = B = L_{v*}^{-1}(B_v^*) \in \mathfrak{h}$.
In other words, the relationship between $\omega_{\mathcal{G}}$ and $\omega_{\mathcal{G}\times_G H}$ is that
$$\omega_{\mathcal{G}}(X_u) = 0\text{ iff }\omega_{\mathcal{G}\times_G H}(\widetilde\varphi_{b*} X_u) = 0. \tag{3}$$
Together with $\omega_{\mathcal{G}\times_G H}(A^*_v) = A$ for $A \in \mathfrak{h}, v \in \mathcal{G} \times_G H$ that fully determines $\omega_{\mathcal{G}\times_G H}$. If we interpret $\omega_{\mathcal{G}\times_G H}$ at a point $v = \overline{(u,b)}$ as the map from $T_u\mathcal{G}\times_{T_{e_G}G} T_bH \to \mathfrak{h}$, then it is given by
$$\begin{multline}\omega_{\mathcal{G}\times_G H}(X_{u}+B_{b}^*) = \text{ad}_{b^{-1}}\left(\varphi_*(\omega_{\mathcal{G}}(X))\right) + B \\= \text{Ad}_{b^{-1}}\left(\varphi_*(\omega_{\mathcal{G}}(X))\right) + L_{b*}^{-1}(B_{b}^*). \tag{4}\end{multline}$$
Here the first component is the original $\omega$ up to the $\text{Ad}$ and $\varphi_*$, and the second is Maurer-Cartan form.
Note that here I use the following notation:
$$\text{ad}_a(b) = aba^{-1} = L_a R_{a^{-1}} b,\quad \text{Ad}_a(B) = (\text{ad}_a)_* B = L_{a*} R_{a^{-1}*} B. \tag{5}$$
This is consistent with Kobayasi, Nomidzu but may differ from other textbooks.
Formula (4) is, perhaps,the formula you were looking for. There are 2 ways to show that $\omega_{\mathcal{G}\times_G H}$ given by (4) is well defined ($TG$-balanced): (1) from construction we derived the formula from the description of the connection, hence (by construction) it is a valid connection form on $T(\mathcal{G} \times_G H)$; (2) one can show it directly. Let's show that directly. Let $A \in \mathfrak{g}$. We have
$$\omega_{\mathcal{G}\times_G H}(A_u^* - R_{b*}\varphi_* A) = \text{Ad}_{b^{-1}} (\varphi_*A) - L_{b^{-1}*} R_{b*} \varphi_* A = 0. \tag{6}$$
