A question very close to this one was already asked: [Automorphisms of a weighted projective space](https://mathoverflow.net/questions/67363/automorphisms-of-a-weighted-projective-space)


But the answer given does not satisfy my needs. So avoiding having two questions that are identical, I am interested in a specific type of weighted projective space, namely $\mathbb{P}(1, 1, \cdots, 1, k)$ for some natural number $k$. The case $k = 1$ gives rise the usual projective space. To be more precise, I am considering a real weighted projective space with weight vector $(1, 1, \cdots, 1, k)$. 


So the question is can one characterize, in general, the automorphism group of such a space. If $k = 1$ then we get the projective linear group, namely the group $\textbf{GL}(\mathbb{R}^n)/\mathbb{R}^\ast \cong \textbf{PGL}(\mathbb{R}^n)$. In general, can one characterize the automorphism group of weighted projective spaces of the type $\mathbb{P}(1, 1, \cdots, 1, k)$?



Automorphisms of a specific type of weighted projective space
I'm not sure how one deals with the general case, but your example is the cone in $\mathbb P^{N+1}$ over the $k$-th Veronese embedding of $\mathbb P^{n-1}(\mathbb K)$ in $\mathbb P^N$ ($n=$ number of 1's, $N=$ the appropriate dimension for the Veronese embedding). 


So for $k>1$ the automorphism group $G$ is an extension 
$$1\to \mathbb K^\*\times {\mathbb K}^{N+1}\to G\to \mathbb PGL(n)\to 1.$$
EDIT: as J\'er\'emy points out below and in his answer, the kernel in the sequence above is not a direct product, but a more complicated group.


This is easily seen by taking homogeneous coordinates in $\mathbb P^{N+1}$ such that the vertex of the cone is, say, $[1,0\dots 0]$ and representing an automorphism by a matrix such that the first column is $^t(1,0\dots 0)$.


