= Solution
First use the nonzero loci, as required by the standard <SL2R action on differentials>. A nonzero <holomorphic one-form> has, away from its zeros, <flat coordinates>
$$
w=\int\omega,\qquad\omega=dw,
$$
whose changes of coordinate are translations. A nonzero <holomorphic quadratic differential> similarly has local <flat coordinates> $w=\int\sqrt q$, with $q=dw^2$ and changes of coordinate $w_j=\pm w_i+c_{ij}$. These are respectively <translation surfaces> and <half-translation surfaces>.
Identify a <flat coordinate> with a vector in $\mathbb R^2$. For $A\in\mathrm{SL}_2(\mathbb R)$, replace every <flat coordinate> by $W=Aw$. Since $A$ preserves <orientation> and commutes with multiplication by $-1$, the new changes of coordinate are
$$
W_j=W_i+Ac_{ij},\qquad\text{or}\qquad W_j=\pm W_i+Ac_{ij}.
$$
They are <holomorphic> in the new coordinates, and so define a new <complex structure>. Define $\omega_A=dW$ or $q_A=dW^2$ in that structure. The forms glue because translations preserve $dW$, and the extra signs preserve $dW^2$.
The zeros also extend. A zero of order $m$ of a <holomorphic one-form> has <cone angle> $2\pi(m+1)$; a zero of order $m$ of a <holomorphic quadratic differential> has <cone angle> $(m+2)\pi$. The real-linear deformation preserves the corresponding winding multiplicity. Filling the cone in a local coordinate $\zeta$ gives $W=\zeta^{m+1}$ in the first case, or a local branch of $W=\zeta^{(m+2)/2}$ in the second. Thus the resulting forms are constant multiples of $\zeta^m\,d\zeta$ or $\zeta^m\,d\zeta^2$ and have the same <zero orders>. This verifies extension across the missing points, rather than merely producing an atlas on the punctured surface.
An isomorphism preserving the original differential identifies its <flat coordinates> up to the permitted translations or signs; applying $A$ identifies the deformed atlases too. Hence the construction descends to the corresponding <moduli spaces>. Applying $B$ after $A$ replaces $w$ by $BAw$, so
$$
\boxed{B\cdot(A\cdot(X,\omega))=(BA)\cdot(X,\omega),\qquad
B\cdot(A\cdot(X,q))=(BA)\cdot(X,q).}
$$
The <area of a quadratic differential>, and the analogous area of a <holomorphic one-form>, are preserved because $\det A=1$.
For $q=\omega^2$, the <flat coordinates> obtained from $\omega$ already give the required <half-translation surface> atlas for $q$. The same replacement $w\mapsto Aw$ therefore constructs both deformations, and
$$
\boxed{A\cdot(X,\omega^2)=(X_A,\omega_A^2)=s\bigl(A\cdot(X,\omega)\bigr).}
$$
The printed sets include identically zero differentials. They have no <flat coordinates>, so the customary geometric <group action> is defined on the nonzero loci. One can obtain a set-theoretic action on the displayed entire sets by declaring $A\cdot(X,0)=(X,0)$; the same equivariance identity then holds at zero. This extension is generally not continuous: as $t\omega\to0$, the deformed underlying surface is the same $X_A$ for every real $t>0$, and can differ from $X$. Thus a claim about the standard continuous geometric <group action> requires the nonzero convention.
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