I have been thinking about quotients lately and pondered the following:

Let $G$ be a connected linear algebraic group and $X$ a $G$-variety acting via the morphism $\sigma:G\times X\rightarrow X$. Let $p:L\rightarrow X$ be a line bundle on $X$.

A $G$-linearisation of $L$ is an action of $G$ on $L$ such that $p(g\cdot l)= g\cdot p(l)$, for $l\in L, g\in G$, and which restricts to a linear isomorphism $L_{x}\rightarrow L_{g\cdot x}$ on the fibres. This last condition can be expressed as saying that there is an isomorphism $L\rightarrow g^{\ast}L$, for each $g\in G$ (here $g^{\ast}L$ is the pullback bundle by the automorphism $g$ of $X$). In fact, since $G$ is connected, a $G$-linearisation of $L$ exists if and only if there is an isomorphism $p_{2}^{\ast}L\rightarrow \sigma^{\ast}L$ of bundles on $G\times X$, with $p_{2}$ the projection to $X$.

It is known (Corollary 7.2, p.109, 'Lectures on Invariant Theory' - Dolgachev) that if $X$ is normal then for any $L$ there is some power of $L$ that admits a $G$-linearisation.

Question 1: Can someone provide an example of a non-normal $G$-variety $X$ and a line bundle $L$ for which no power $L^{n}$ admits a $G$-linearisation? .

Question 1': If no such example can exist can someone point me towards the literature (if any) where this question is addressed?

The existence result for normal $X$ relies on that fact that there is an exact sequence

$ 0\rightarrow K \rightarrow Pic^{G}(X)\rightarrow Pic(X) \rightarrow Pic(G)$

and that $Pic(G)$ is finite. Here $K$ is the group of rational characters of $G$ and $Pic^{G}(X)$ is the group of line bundles admitting a $G$-linearisation (or line $G$-bundles in Dolgachev's terminology).

Question 2: Can we extend the exact sequence

$0\rightarrow K \rightarrow Pic^{G}(X) \rightarrow Pic(X)$

to the right for arbitrary $X$ and in a 'canonical' manner? (ie, is this exact sequence the tail of a canonical long exact sequence for any $G$-variety X?)

Question 2': If so, what groups appear? Do they have any 'down-to-earth' interpretations? (eg, we have $Pic(G)$ appearing for normal $X$).

Thanks in advance and apologies if this is standard material in GIT - I only have a copy of Dolgachev's notes at hand and these questions are not addressed.