By using agetty

How

Install agetty:

# apk add agetty

Edit /etc/inittab to use agetty
Example for the virtual terminal tty1:
tty1::respawn:/sbin/agetty --autologin root tty1 linux

Example inittab entry for a serial terminal on ttys01:
ttyS0::respawn:/sbin/agetty --autologin root ttyS0 vt100

Tip: You can change the `tty1` or `ttyS0` to a different serial port or virtual terminal as you please. `root` can be changed to a different user as well. Finally the terminal type (`linux` and `vt100` in our examples) can be changed to a wide variety of serial terminals.

By making your own autologin wrapper

How

Writing a wrapper script, called autologin, around /bin/login and moving it in /usr/sbin/
Editing /etc/inittab specifying the use of /usr/sbin/autologin instead of /bin/login

Writing the autologin script

Edit and create the script in this example in the location /usr/sbin/autologin:

Contents of /usr/sbin/autologin

#!/bin/sh
exec login -f root

Remember to make the scripts executable:

# chmod +x /usr/sbin/autologin

The script executes the login binary (part of busybox) which will be searched in $PATH.

As parameters are passed:

-f flag which stands for "Do not authenticate (user already authenticated)"
username in this example is root but if you created a new user, its username can be used instead.

Editing /etc/inittab

Open /etc/inittab

replace ":respawn:/sbin/getty" with ":respawn:/sbin/getty -n -l /usr/sbin/autologin" for each TTY you want to enable autologin.

The getty's -n flag do not prompt the user for a login name
The getty's -l flag invokes a custom login instead of /bin/login; in our case it is set to invoke /usr/sbin/autologin

Note

To perform such a replacement on all TTYs, the following command can be used:

# sed -i 's@:respawn:/sbin/getty@:respawn:/sbin/getty -n -l /usr/sbin/autologin@g' /etc/inittab

"@" is used as a delimiter
The -i flag edits the file in-place

References