Big company X provides its employees with Android phones. Those employees don't necessarily own a Play Store account (gmail address required AFAIK).
At the moment, my alternative is to send a link to our .apk file, however the installation procedure might be quite complicated, not even mentioning the updates.
What's the best way to deploy my app to these phones?
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Uchiha Madara is right, but it's not very convenient. They can also subscribe to Google Apps for Business ! Besides, if phones are Lollipop phones, you can very easily enable Android For Work, with pretty useful features to controll mass-hardware !
However, it's not the unique solution. You can self-host your apk file as you do, make a webservice and make your app able to communicate with to check updates ! The only problem is to teach users how to allow install from unknown sources (in security settings), which may be hard if they aren't very confortable with Android phones.
Other solution, some companies offer whole turnkey solutions, like PushLink
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My guess is you will have to use DropBox and they can download it as long as there is Wi Fi of course and ro to the settings > security and check Unknown sources so that it can install the apk from the internet, they can then go to their downloads folder and see the apk there, then install it onto their phones. Hope this helped!
Source: http://www.cnet.com/how-to/how-to-install-apps-outside-of-google-play/
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I think the best way to achieve your requirements is using Microsoft Intune.
Microsoft Intune is a cloud-based service that focuses on mobile device management (MDM) and mobile application management (MAM). You control how your organization’s devices are used, including mobile phones, tablets, and laptops. You can also configure specific policies to control applications. For example, you can prevent emails from being sent to people outside your organization. Intune also allows people in your organization to use their personal devices for school or work. On personal devices, Intune helps make sure your organization's data stays protected and can isolate organization data from personal data.
You could try HockeyApp to allow the users to install the application and know them about the updates. http://hockeyapp.net/features/