A large percentage of the completed suicides in women in the UK are by people with a diagnosis of schizophrenia

About 20% of a the direct victims of suicide in the UK are by people

with a diagnosis of schizophrenia. About 50% of the suicides are from
men and 35% from women (don't know what happened to the other 15% - it
can't be that hard to record gender after a suicide).(Mortensen and Juel
(1993) cited in http://pb.rcpsych.org/cgi/content/full/25/2/46). There
are several other sources that note the considerably higher risk of
completed suicide of women with a diagnosis of schizophrenia compared to
the national average for women. The ration for the whole population is
something like 75% of completed suicides are men.

It has been noted that the higher rate in the general population is
because men choose more violent or effective methods. I'm not 100% sure
this is true.

If the effectiveness of choice of method for suicide was a general
gender constant then it would show in schizophrenia suicides by gender
data. It does and it doesn't. There's clearly other factors in
successful suicide other than choice of method. I am not considering
data on suicide attempts.

I don't know enough about the suicides to make a conclusion.