In the code, I see that ExtensionRefreshException thrown in Jenkins.refreshExtensions() is handled to raise IOException2 and fail plugin installation process.
So it seems that such behavior is by design? If so, then I'd suggest to change this behavior, and this bug report should be reclassified as feature request.

@Gavriil
And so this plugin currently says that its extension is not refreshable and you hope that it can be installed without restarting?
I would say that something like this, saying that it can not be refreshed and so that it can not be installed without restarting is expected.
In fact, if this plugin and the update-center can not do that, they should certainly say it.

What do you expect instead?

Hi evernat,

A year has passed, but as I remember the issue – is that I'd try to install other plugin not this. And installation process failed due to non-refreshable extension was activated.

So I expect that installation of one plugin shouldn't be affected with other extension.

I understand your point of view now. It makes sense.

But I think that the best solution is that this Persona plugin just upgrades its parent Jenkins dependency.
And so that a "refresh()" method is added in hudson.plugins.persona.xml.XmlPersonaFinder
It would make it compatible with installing a plugin without restarting.

Anyway, this issue is probably valid and needs to be fixed one way or the other.
(But I am not the author of the persona plugin)

Code changed in jenkins
User: ssogabe
Path:
pom.xml
src/main/java/hudson/plugins/persona/random/RandomPersonaFinder.java
src/main/java/hudson/plugins/persona/xml/XmlPersonaFinder.java
http://jenkins-ci.org/commit/persona-plugin/19321630fe9c0cccbc5a670baac4b7b1415807e4
Log:
[FIXED JENKINS-12803] support installing plugin without restarting

Compare: https://github.com/jenkinsci/persona-plugin/compare/601e38a4c03c...19321630fe9c

Still an issue for me with persona-2.3..

hudson.util.IOException2: Failed to dynamically deploy this plugin
at hudson.model.UpdateCenter$InstallationJob._run(UpdateCenter.java:1317)
at hudson.model.UpdateCenter$DownloadJob.run(UpdateCenter.java:1116)
at java.util.concurrent.Executors$RunnableAdapter.call(Executors.java:471)
at java.util.concurrent.FutureTask$Sync.innerRun(FutureTask.java:334)
at java.util.concurrent.FutureTask.run(FutureTask.java:166)
at hudson.remoting.AtmostOneThreadExecutor$Worker.run(AtmostOneThreadExecutor.java:104)
at java.lang.Thread.run(Thread.java:722)
Caused by: hudson.util.IOException2: Failed to install jeff-plugin plugin
at hudson.PluginManager.dynamicLoad(PluginManager.java:438)
at hudson.model.UpdateCenter$InstallationJob._run(UpdateCenter.java:1313)
... 6 more
Caused by: java.lang.NullPointerException
at java.util.ArrayList.batchRemove(ArrayList.java:669)
at java.util.ArrayList.removeAll(ArrayList.java:640)
at hudson.plugins.persona.xml.XmlPersonaFinder.refresh(XmlPersonaFinder.java:87)
at jenkins.model.Jenkins.refreshExtensions(Jenkins.java:2138)
at hudson.PluginManager.dynamicLoad(PluginManager.java:431)
... 7 more