There is such a configuration:
Internet ===> [WAN port] Grandstream HT812 with hosted PBX users [FXS ports] ===> analog ministation Panasonic ===> analog telephone sets
Situation: When a caller dials a number which belongs to a hosted PBX, the call lands on the HT812, then it is switched to the ministation. While an IVR of the ministation is playing, the caller presses some additional internal numbers. After that he is switched to a telephone set with the corresponding internal number.
Problem: When dialing internal numbers, callers are not always switched to the corresponding telephone set. In these cases they are switched to the default telephone set (which belongs to a secretary), as if a wrong number was dialed. When dialing from IP phones, pressing additional numbers almost always works fine, but when dialing from mobile phones, it almost always fails.
Prehistory: Earlier instead of Grandstream HT812 was Cisco SPA122. First the problem with dialing internal numbers was the same. A sysadmin who previously ran this system had changed some settings on SPA122 on the advice of other people, which solved the problem. But, alas, he doesn't remember what he had done exactly. He only remembers that he changed DTMF method: he turned off 'RFC2833' and switched to 'Inband' or 'SIP INFO'.
Question 1: According to theory, "DTMF (Dual Tone Multi-frequency), better known as touch-tone, is used for telecommunication signaling over analog telephone lines in the voice-frequency band. DTMF is the signal sent from the IP phone to the network, which is generated when pressing the IP phone's keypad during a call." If DTMF is sent from an IP phone (here Cisco ATA) to the network (here Internet), how could it affect dialing internal numbers by callers, which are sent in the opposite direction--from ATA to ministation?
Question 2: What should I change on the Grandstream HT812 to make internal numbers work fine? I've tried to simply changed preferred DTMF method to 'In-audio', but it didn't help. Here are some relevant settings on HT812 (square brackets mark current settings):
DTMF Payload Type: 
Preferred DTMF method: [RFC2833] / In-audio / SIP INFO
Inband DTMF Duration (in 40-2000 ms range): duration: , inter-duration: 
Disable DTMF Negotiation: [No] (negotiate with peer) / Yes (use above DTMF order without negotiation)
Generate Continuous RFC2833 Events: [No] / Yes (RFC2833 events are generated until key is released)