Are you planning to do this, Peter? This is a critical part of the dev preview. I can take a stab if you want.

IIUC at least on some versions of Android the package installer should prompt you to flip if it's not set to allow install. So this might just be "send it off to the installer"?

I will take this on

The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/74d273a69f5e0954c0aa84a82fa5dfc55c3754d4

commit 74d273a69f5e0954c0aa84a82fa5dfc55c3754d4
Author: pkotwicz <pkotwicz@chromium.org>
Date: Wed Aug 10 20:13:57 2016

Install downloaded WebAPK when "installation from unknown sources" is enabled

BUG= 635373 

Review-Url: https://codereview.chromium.org/2231803003
Cr-Commit-Position: refs/heads/master@{#411128}

[modify] https://crrev.com/74d273a69f5e0954c0aa84a82fa5dfc55c3754d4/chrome/android/java/src/org/chromium/chrome/browser/ChromeApplication.java
[modify] https://crrev.com/74d273a69f5e0954c0aa84a82fa5dfc55c3754d4/chrome/android/java/src/org/chromium/chrome/browser/webapps/WebApkInstaller.java

The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/f5345736320415ba85e7543bd38f4e8aa897cdf0

commit f5345736320415ba85e7543bd38f4e8aa897cdf0
Author: pkotwicz <pkotwicz@chromium.org>
Date: Fri Aug 12 21:58:45 2016

Make downloaded WebAPK world readable

Make downloaded WebAPK world readable. This is necessary in order to install
WebAPKs when the user has enabled "installation from unknown sources".

BUG= 635373 

Review-Url: https://codereview.chromium.org/2236033003
Cr-Commit-Position: refs/heads/master@{#411791}

[modify] https://crrev.com/f5345736320415ba85e7543bd38f4e8aa897cdf0/chrome/android/java/src/org/chromium/chrome/browser/webapps/WebApkInstaller.java
[modify] https://crrev.com/f5345736320415ba85e7543bd38f4e8aa897cdf0/chrome/browser/android/webapk/webapk_installer.cc
[modify] https://crrev.com/f5345736320415ba85e7543bd38f4e8aa897cdf0/chrome/browser/android/webapk/webapk_installer.h