Consider $E$ a rank $n$ real vector bundle over $X$. Let $P_O(E)$ be the orthonormal frames of $E$. This is a principal bundle $P_O(E)\to X$ with $O(n)$ action. Thus there is $i:O(n)\to P_O(E),\pi:P_O(E)\to X$ where $i$ is any inclusion fiber map.
"there is an exact sequence, $0\to H^0(X,Z_2)\to H^0(P_O(E),Z_2)\to H^0(O(n),Z_2)\to H^1(X,Z_2)$."
$\textbf{Q:}$ Where is this Cech cohomology exact sequence coming from? My guess is that first map is given by pulling back $0-$th cocycle from $X$ to $P_O(E)$. The second map is given by inclusion. My guess is that this cohomology level map is insensitive to homotopy. Thus I take any fiber inclusion will induce the second map. Is this correct? Any reference indicating Cech cohomology invariant under homotopy of maps?
$\textbf{Q':}$ Where is last map coming from(i.e. $H^0(O(n),Z_2)\to H^1(X,Z_2)$)? Normally, to induce long exact sequence of sheaves of abelian groups, one needs short exact sequence of 3 sheaves of abelian groups. My guess is that one take the inclusion map's cocycle by noting that fiber is closed in $P_O(E)$. Then one can try to extend it globally to $P_O(E)$ and then taking coboundary operator produces an element in $H^1(P_O(E),Z_2)$ but this element may not descend to $H^1(X,Z_2)$.
Ref. Spin Geometry, Lawson, Chpt 2, (1.2)