Why was the Austro-Hungarian government not really concerned whether Serbia itself was involved in Archduke Ferdinand's assassination?

The reason for this is that the government of Austria-Hungary
really wanted a war with Serbia. They did not really care what the truth was. Instead, they
wanted a way to force Serbia to go to war with them.

Austria-Hungary
wanted Serbia destroyed or seriously hurt for two major reasons. First, the presence of Serbia
was serving to inflame pan-Slavic nationalism among Austria-Hungary's Slavic subjects. Serbia was
helping to fan these flames and was even thinking about trying to take Bosnia (a Slavic region of
the empire) away from Austria-Hungary. Secondly, Austria-Hungary wanted more territory in the
Balkans and it specifically wanted to prevent Serbia from gaining a port on the Adriatic.
Austria-Hungary had become more and more worried about both of these issues in the wake of the
Second Balkan War.

For these reasons, Austria-Hungary was eager to
go to war with Serbia. That meant that they would not really care about the truth and would take
any reason they could find to start a war.