Getting new files to inherit group permissions on Linux

It sounds like you're describing the setgid bit functionality where when a directory that has it set, will force any new files created within it to have their group set to the same group that's set on the parent directory.

Example

$ whoami
saml

$ groups
saml wheel wireshark

setup a directory with perms + ownerships

$ sudo mkdir --mode=u+rwx,g+rs,g-w,o-rwx somedir
$ sudo chown saml.apache somedir
$ ll -d somedir/
drwxr-s---. 2 saml apache 4096 Feb 17 20:10 somedir/

touch a file as saml in this dir

$ whoami
saml

$ touch somedir/afile
$ ll somedir/afile 
-rw-rw-r--. 1 saml apache 0 Feb 17 20:11 somedir/afile

This will give you approximately what it sounds like you want. If you truly want exactly what you've described though, I think you'll need to resort to Access Control Lists functionality to get that (ACLs).

ACLs

If you want to get a bit more control over the permissions on the files that get created under the directory, somedir, you can add the following ACL rule to set the default permissions like so.

before

$ ll -d somedir
drwxr-s---. 2 saml apache 4096 Feb 17 20:46 somedir

set permissions

$ sudo setfacl -Rdm g:apache:rx somedir
$ ll -d somedir/
drwxr-s---+ 2 saml apache 4096 Feb 17 20:46 somedir/

Notice the + at the end, that means this directory has ACLs applied to it.

$ getfacl somedir
# file: somedir
# owner: saml
# group: apache
# flags: -s-
user::rwx
group::r-x
other::---
default:user::rwx
default:group::r-x
default:group:apache:r-x
default:mask::r-x
default:other::---

after

$ touch somedir/afile
$ ll somedir/afile 
-rw-r-----+ 1 saml apache 0 Feb 17 21:27 somedir/afile
$ 

$ getfacl somedir/afile
# file: somedir/afile
# owner: saml
# group: apache
user::rw-
group::r-x              #effective:r--
group:apache:r-x        #effective:r--
mask::r--
other::---

Notice with the default permissions (setfacl -Rdm) set so that the permissions are (r-x) by default (g:apache:rx). This forces any new files to only have their r bit enabled.

TL:DR; to make new files inherit the group of the container folder do:

$ chmod g+s somefolder

Note: its implied in the accepted answer, this is just a snippet.

As a complement to slm's answer, note that, on an ext2/3/4 filesystem, you can replicate the BSD behavior you describe by using the bsdgroups mount option on the partition. From the mount(1) man page :

grpid|bsdgroups and nogrpid|sysvgroups
              These options define what group id a newly  created  file  gets.
              When  grpid  is  set,  it takes the group id of the directory in
              which it is created; otherwise (the default) it takes the  fsgid
              of  the current process, unless the directory has the setgid bit
              set, in which case it takes the gid from the  parent  directory,
              and also gets the setgid bit set if it is a directory itself.