Is this the official Cloud SQL image? Are extensions supported?

[Fohlen]

According to the documentation Cloud SQL supports a lot of extensions.

When accessing a new instance of marketplace.gcr.io/google/postgresql15 like so:

\dx

I get:

List of installed extensions
  Name   | Version |   Schema   |         Description          
---------+---------+------------+------------------------------
 plpgsql | 1.0     | pg_catalog | PL/pgSQL procedural language
(1 row)

Which is fine. However, if I try to create e.g pgvector:

postgres=# CREATE EXTENSION vector;
ERROR:  extension "vector" is not available
DETAIL:  Could not open extension control file "/usr/share/postgresql/15/extension/vector.control": No such file or directory.
HINT:  The extension must first be installed on the system where PostgreSQL is running.

According to the documentation, it is.
Is it, or is it not supported? If it is, what are the steps to make it work?
If it is not, why is it listed as supported. Is this not the official image deployed in the Cloud SQL product?
This is rather misleading.