Let $M$ be an $n$ dimensional manifold with its bundle of linear frames $\pi:L(M)\to M$. Suppose $T_0$ is a tensor on $\mathbb R^n$ and $u\in L(M)$. We may view $u$ as a linear map $u:\mathbb R^n\to T_{\pi(u)}M$. Obtain the induced morphism of tensor algebras $u_*:\mathfrak T(\mathbb R^n)\to\mathfrak T(T_{\pi(u)}M)$.
How may we view $T=u_*T_0$ as a section of $\mathfrak T(M)$, when $T$ is only a tensor above $\pi(u)$?
Now let $G<GL(n,\mathbb R)$ be the largest Lie subgroup that leaves $T_0$ invariant. How can we use invariance of $T_0$ to define a section of the associated bundle $L(M)/G$?
We can't. Hovever by changing $u$ in $L(M)$ we can. In case of $G$ structure wehere $G$ is a group of transformations leaving $T_0$ untached wewill get a tensor on $M$. Which leads to the answear to the second question.
Namely we have a corespondance between $G$ structures and tensor fields $0$-reductible to $T_0$. When having $G$ structure $P \subset L(M)$ we define $$T_x={u_x}_* T_0$$ when $u_x \in P_x$ and the definition is correct since any two $u_x$'s in $P_x$ differ by an element of $G$. When having $T$ we simply define $$ P_x = \left\{ u_x \in L(M)_x :{u_x}_*T_0=T_x \right\} .$$