Let $f: S_A \to S_B$ be a homomorphism from the symmetric group on $A$ to the symmetric group on $B$, where $A$ and $B$ may be infinite. For $X\subseteq A$ and $b_1,b_2\in B$, say that $b_1\sim_X b_2$ if and only if $f(g)(b_1)=b_2$ for some $g$ s.t. $g(x) = x$ for all $x\in X$. Define $h: \mathcal{P}(B)\to\mathcal{P}(\mathcal{P}(A))$ s.t. $h(Y)=\{X: Y\text{ is closed under }\sim_X\}$. Call $Y$ principal if $\bigcap h(Y)\in h(Y)$ and boring if $\emptyset\in h(Y)$.
Question 1: Is it possible that every principal $Y\subseteq B$ be boring but not every $Y\subseteq B$ be boring?
Question 2: More generally, could there be a complete proper subalgebra of the power set algebra $\langle\mathcal{P}(B),\cap,\cup\rangle$ that contains all principal elements of $\mathcal{P}(B)$? ("More generally" because an affirmative answer to Question 1 implies that the boring subsets of $B$ form such a subalgebra.)
Question 3: Is there standard terminology for and/or standard results about the notions defined above?
I think the answer to Question 1 (and hence Question 2) is "yes", unless I've made a mistake.
For $X,Y\in\mathcal{P}(\mathbb{N})$ and $g\in\mathbb{N}^\mathbb{N}$, let $g(X) = \{g(x): x\in X\}$, $E = \{\langle X,Y\rangle: |(X\cup Y)\backslash(X\cap Y)|<\omega\}$, and $[X] = \{Y: \langle X,Y\rangle\in E\}$. Now let $A = \mathbb{N}$ and $B=(\mathcal{P}(\mathbb{N})/E)\backslash\{[\emptyset],[\mathbb{N}]\}$, and define $f: S_A\to S_B$ s.t. $f(g)([X]) = [g(X)]$. By construction, $h(Z)$ must be closed under $E$ for arbitrary $Z\subseteq B$; hence $Z$ is principal only if it is boring. And not every $Z$ is boring, since only $\emptyset$ and $B$ are boring.