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The equality follows directly from the definition of a connection, and is independent of the context of lifting structure groups, or degree three cohomology.

Recall that a connection on a principal $G$-bundle $\pi:P \to M$ is a 1-form $\omega \in \Omega^1(P,\mathfrak{g})$ such that $$ p_2^{\ast}\omega = Ad_g^{-1} (p_1^{\ast}\omega) + g^{\ast}\theta $$ over $P \times_M P$, where $p_1,p_2: P \times_M P \to P$ are the two projections, and $g: P \times_M P \to G$ is the difference map defined by $g(p,p')\cdot p=p'$. Here $g^\ast\theta=g^{-1}dg$, whatever notation is preferred.

Now, if $b: P \to G$ is a smooth map, it induces a map $$ \tilde b: P \to P \times_M P: p \mapsto (p\cdot f(p),p). $$ Pullback of above defining equation along $\tilde b$ produces $$ \omega = Ad_f(f_{\ast}\omega)+f^*\theta. $$ If $b$ is central, this is your equation.

This works of course also, if $b:M \to G$ is defined on the base (you didn't say clearly where $b$ is defined). Then use $b' := b \circ \pi$ instead.

The equality follows directly from the definition of a connection, and is independent of the context of lifting structure groups, or degree three cohomology.

Recall that a connection on a principal $G$-bundle $\pi:P \to M$ is a 1-form $\omega \in \Omega^1(P,\mathfrak{g})$ such that $$ p_2^{\ast}\omega = Ad_g^{-1} (p_1^{\ast}\omega) + g^{\ast}\theta $$ over $P \times_M P$, where $p_1,p_2: P \times_M P \to P$ are the two projections, and $g: P \times_M P \to G$ is the difference map defined by $g(p,p')\cdot p=p'$. Here $g^\ast\theta=g^{-1}dg$, whatever notation is preferred.

Now, if $b: P \to G$ is a smooth map, it induces a map $$ \tilde b: P \to P \times_M P: p \mapsto (p\cdot b(p),p). $$ Pullback of above defining equation along $\tilde b$ produces $$ \omega = Ad_b(b_{\ast}\omega)+b^*\theta. $$ If $b$ is central, this is your equation.

This works of course also, if $b:M \to G$ is defined on the base (you didn't say clearly where $b$ is defined). Then use $b' := b \circ \pi$ instead.

The equality follows directly from the definition of a connection, and is independent of the context of lifting structure groups.

Recall that a connection on a principal $G$-bundle $\pi:P \to M$ is a 1-form $\omega \in \Omega^1(P,\mathfrak{g})$ such that $$ p_2^{\ast}\omega = Ad_g^{-1} (p_1^{\ast}\omega) + g^{\ast}\theta $$ over $P \times_M P$, where $p_1,p_2: P \times_M P \to P$ are the two projections, and $g: P \times_M P \to G$ is the difference map defined by $g(p,p')\cdot p=p'$. Here $g^\ast\theta=g^{-1}dg$, whatever notation is preferred.

Now, if $b: P \to G$ is a smooth map, it induces a map $$ \tilde b: P \to P \times_M P: p \mapsto (p\cdot f(p),p). $$ Pullback of above defining equation along $\tilde b$ produces $$ \omega = Ad_f(f_{\ast}\omega)+f^*\theta. $$ If $b$ is central, this is your equation.

This works of course also, if $b:M \to G$ is defined on the base (you didn't say clearly where $b$ is defined). Then use $b' := b \circ \pi$ instead.

The equality follows directly from the definition of a connection, and is independent of the context of lifting structure groups, or degree three cohomology.

Recall that a connection on a principal $G$-bundle $\pi:P \to M$ is a 1-form $\omega \in \Omega^1(P,\mathfrak{g})$ such that $$ p_2^{\ast}\omega = Ad_g^{-1} (p_1^{\ast}\omega) + g^{\ast}\theta $$ over $P \times_M P$, where $p_1,p_2: P \times_M P \to P$ are the two projections, and $g: P \times_M P \to G$ is the difference map defined by $g(p,p')\cdot p=p'$. Here $g^\ast\theta=g^{-1}dg$, whatever notation is preferred.

Now, if $b: P \to G$ is a smooth map, it induces a map $$ \tilde b: P \to P \times_M P: p \mapsto (p\cdot f(p),p). $$ Pullback of above defining equation along $\tilde b$ produces $$ \omega = Ad_f(f_{\ast}\omega)+f^*\theta. $$ If $b$ is central, this is your equation.

This works of course also, if $b:M \to G$ is defined on the base (you didn't say clearly where $b$ is defined). Then use $b' := b \circ \pi$ instead.

The equality follows directly from the definition of a connection, and has nothing to do with this lifting stuff.

Recall that a connection on a principal $G$-bundle $\pi:P \to M$ is a 1-form $\omega \in \Omega^1(P,\mathfrak{g})$ such that $$ p_2^{\ast}\omega = Ad_g^{-1} (p_1^{\ast}\omega) + g^{\ast}\theta $$ over $P \times_M P$, where $p_1,p_2: P \times_M P \to P$ are the two projections, and $g: P \times_M P \to G$ is the difference map defined by $g(p,p')\cdot p=p'$. Here $g^\ast\theta=g^{-1}dg$, whatever notation is preferred.

Now, if $b: P \to G$ is a smooth map, it induces a map $$ \tilde b: P \to P \times_M P: p \mapsto (p\cdot f(p),p). $$ Pullback of above defining equation along $\tilde b$ produces $$ \omega = Ad_f(f_{\ast}\omega)+f^*\theta. $$ If $b$ is central, this is your equation.

This works of course also, if $b:M \to G$ is defined on the base (which you didn't say clearly). Then use $b' := b \circ \pi$ instead.

The equality follows directly from the definition of a connection, and is independent of the context of lifting structure groups.

Recall that a connection on a principal $G$-bundle $\pi:P \to M$ is a 1-form $\omega \in \Omega^1(P,\mathfrak{g})$ such that $$ p_2^{\ast}\omega = Ad_g^{-1} (p_1^{\ast}\omega) + g^{\ast}\theta $$ over $P \times_M P$, where $p_1,p_2: P \times_M P \to P$ are the two projections, and $g: P \times_M P \to G$ is the difference map defined by $g(p,p')\cdot p=p'$. Here $g^\ast\theta=g^{-1}dg$, whatever notation is preferred.

Now, if $b: P \to G$ is a smooth map, it induces a map $$ \tilde b: P \to P \times_M P: p \mapsto (p\cdot f(p),p). $$ Pullback of above defining equation along $\tilde b$ produces $$ \omega = Ad_f(f_{\ast}\omega)+f^*\theta. $$ If $b$ is central, this is your equation.

This works of course also, if $b:M \to G$ is defined on the base (you didn't say clearly where $b$ is defined). Then use $b' := b \circ \pi$ instead.