There's a family of amplifiers of arbitrary gain, of inverting sign, when common-emitter or common-base types are considered.
For the CE case, there are two common approaches, or any combination thereof:
If we have a resistor divider (in the AC equivalent circuit) from collector to base to input, we have shunt feedback, which reduces the input and output impedance, along with gain and distortion. In the high-gain limit (that is, gm or hFE --> infty), the overall gain equals the ratio of resistances.
If we have a resistor divider the transistor is a part of, i.e. some resistance below the emitter (emitter degeneration resistance), and above the collector (load resistance, a parallel combination including any actual load), then the input and output impedances are increased, and gain and distortion are decreased. In the high-gain limit, overall gain equals the ratio of resistances.
The impedance character of these two methods (shunt and series feedback) is important for any given application. Indeed, sometimes the shunt feedback case of equal resistances (thus giving nominal gain = -1) is sometimes called an "inverting follower", in analogy to the emitter follower, but any gain is possible, which seems an adequate example for the question.
For common-base, feedback can be delivered to the base anyway, giving a hybrid approach. In this case, the input impedance is still fairly low (it's into the emitter, plus series degeneration resistance if applicable, or raised also by the effect of base feedback). Alternately, a fraction of output voltage can be fed back in parallel, series, or some combination thereof, by transformer action (sometimes applicable for low-noise RF amplifiers).
The same is of course true of common-collector amplifiers, where some output can be fed back to the base or input, to partially cancel the "100%" degeneration of the emitter follower configuration.
The main problem with the latter two kinds of feedback, is the requirement of transformer coupling, because signal must be subtracted, i.e. the transformer has to invert the feedback signal. So a mere resistor divider cannot provide the service, as can for the CE amplifier (which is inverting by itself).
If we allow multiple transistors, we can consider circuits like the cascode, diff pair, Darlington, etc. which have different properties.
Tim